ms mean or CLUSTER ‘sm The“: $09 Hm Deans of PI». D. MICHIGAN ESTATE UNIVERSITY Ruth Am: Sn ' 1974‘ V {6; -. V uh Mld‘ugsfi? 13923113 ”"" ' :me .‘v: .' ’. - .JV‘J‘EE‘ This is to certify that‘the' ~' ' V ' a‘. . v 1 d . , .. V - ‘ theSIS entlt e ‘5“? ‘ THE THEORY OF CLUSTER SETS presented by Ruth Ann Su has been accepted towards fulfillment of the requirements for Ph.D. Mathematics degree in ffié /// Z3”. ,. Major professor I. T @m W /?‘Z /7>?‘ ”fir. . . V. ‘fi‘ 0-7 639 ___-|-I-'I-I_ » I .n .I.II M23 Since mathematn t0 the 39.1 of defmiI three may Functions Arbi- to chm: SPhere. ] any Compl. countable exist mo Sets, A1 QlUStel. S internau: ABSTRACT THE THEORY OF CLUSTER SETS By Ruth Ann Su Since Painleve founded the theory of cluster sets in 1895, mathematicians have discovered many significant properties pertaining to the set of limit points of a function at the boundary of its domain of definition. The functions studied may be divided into the following three major classes: Arbitrary Functions, Normal Functions and Class A Functions. Arbitrary functions have limited patterns of behavior with respect to cluster sets because of the particular topologies of the plane and SPhere. For example, according to the Bagemihl Ambiguous Point Theorem, any Complex-valued function defined in the unit disk D has at most a countable number of boundary points eie with the property that there eXist two curves in D ending at e19 along which f has disjoint cluster Sets. Also, globally, there are numerous relationships between the Cluster set of a function relative to an angle at a point ei and the Cluster set of a function relative to a region between two circles each i6 internally tangent to the unit circle at e . A function arbitrary confo that every seqr verges uniform? of this region omits at least is normal if i bolic metric t analytic norm; type of normaf the sum of my definition of Of a normal f Placed by the SUppose limes to c151i Bending at . if and Only at ei9 along bonnded by S than or eque which the Inc maximum diar set of 2 he that a film: Ruth Ann Su A function is normal in a simply connected region if its family of arbitrary conformal mappings of the region onto itself has the property that every sequence of this family contains a subsequence which con- verges uniformly or tends uniformly to infinity on every compact subset of this region. A meromorphic function in D is normal if the function omits at least three points in D. In addition a complex function in D is normal if it is uniformly continuous from the disk with the hyper- bolic metric to the sphere with the chordal metric. The sum of two analytic normal functions is not necessarily normal although the special type of normal functions called uniformly normal has the property that the sum of two uniformly normal functions is uniformly normal. The definition of a uniformly normal function is analogous to the definition of a normal function where the sphere with the chordal metric is re- placed by the plane With the usual metric. Suppose f is a holomorphic nonconstant function in D. Then f be— longs to Class A if for each point in a dense set of C, f has a path in D ending at eie along which E approaches a limit. f belongs to Class B if and only if the set of points eie, for which f has a path in D ending at ei6 along which either f approaches infinity or the modulus of f is bounded by some finite number, is dense on C. For any constant k greater than or equal to zero the level set consists of all points z in D for which the modulus of f is equal to X. Then f belongs to Class L if the maximum diameter of the components of each level set intersected with the Set of 2 having modulus greater than r approaches zero as r approaches one. A very important theorem in the study of Class A Functions states that a function is in Class A if and only if the function is in Class B if and only if th closed under the every nonconstant or as the product Ruth Ann Su if and only if the function is in Class L. Class A Functions are not closed under the operations of addition and multiplication. In fact every nonconstant, holomorphic function in D can be written as the sum or as the product of pairs of functions in Class A. in pa THE THEORY OF CLUSTER SETS By Ruth Ann Su A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1974 ©Copyright by RUTH ANN SU 1974 To Lawrence, Billy, John, and Larry ii Iwish to e: for his continua Wish to thank my for the typing 0 ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professor Peter Lappan for his continual assistance in the preparation of this thesis. I also wish to thank my husband Dr Lawrence Su for his helpful suggestions and for the typing of the thesis. iii PREFACE 1 CHAPTER I. ARBI' INTRODUCTION 9 RESULTS RELATED ' RESULTS 0N BOUND SOME SPECIAL Typ HORDCYCLES 28 ORICYCLIC CLUSTE SELECTOR 0F ARCS limoREMS FOR sou Continuous Fun M'Topology for Light Interior Locally Unival Hohmilhic Fu CHAPTER 11. Not SUFFICIENT mum CLUSTER-SET THEC SUBHi‘Ri‘DNIC Nom mRDCYCLIC PROP! TABLE gr; CONTENTS Page PREFACE 1 CHAPTER I. ARBITRARY FUNCTIONS 9 INTRODUCTION 9 . RESULTS RELATED TO BAGEMIHL'S AMBIGUOUS-POINT THEOREM 13 V A RESULTS ON BOUNDARY FUNCTIONS 17 SOME SPECIAL TYPES OF CLUSTER SETS 24 HORDCYCLES 28 ORICYCLIC CLUSTER SETS 40 SELECTOR OF ARCS 44 THEOREMS FOR SPECIAL TYPES OF FUNCTIONS 52 Continuous Functions 52 M-Topology for Continuous Functions 57 Light Interior Functions 60 Locally Univalent Functions 62 Holomorphic Functions 64 Lisa‘s-..‘ . K i. CHAPTER II. NORMAL FUNCTIONS 69 SUFFICIENT CONDITIONS FOR A FUNCTION TO BE NORMAL 69 CLUSTER-SET THFDREMS FOR NORMAL FUNCTIONS 79 SUBHARMONIC NORMAL FUNCTIONS 89 BOUNDARY BEHAVIOR OF NORMAL FUNCTIONS 92 HOROCYCLIC PROPERTIES OF I‘DRMAL FUNCTIONS 99 A FUNCTION THEORETIC CHARACTERIZATION OF NORMAL MEROMORPHIC FUNCTIONS 102 NORMAL HOIDMORPHIC FUNCTIONS 106 NORMAL HARMONIC FUNCTIONS 110 CHAPTER III. CLASS A FUNCTIONS 113 INTRODUCTION 113 PROPERTIES OF CLASS A FUNCTIONS 114 SUFFICIENT (DNDITIONS FOR f E A 115 BARTH'S GENERALIZATIONS 0F MACLANE'S RESULTS 117 ALGEBRAIC OPERATIONS OF CLASS A FUNCTIONS 125 CALCULUS PROPERTIES OF CLASS A FUNCTIONS 130 BIBLIOGRAPHY 141 GENERAL REFERENCES 149 RECENT REFERENCES APPLICABLE TO FUTURE RESEARCH 156 iv The purpose developments in would have the or course in the th the works by Col Painleve (1 gave the name “d aiunction at a called the clust was first consid general function topological . Early devel corned with the an isolated esst larities. The 1 proved in a pep an isolated poi in D - B, then Paint or the en theorem is true If E is a] PREFACE The purpose of this thesis is to bring together recent important developments in the theory of cluster sets. We assume that the reader would have the mathematical training equivalent to that of a graduate course in the theory of complex variables and a cursory knowledge of the works by Collingwood and Lohwater (l), Noshiro (l) and MacLane(l). Painleve (1) founded the theory of cluster sets in 1895 when he gave the name ”domains d'indetermination" to the set of limit points of a function at a boundary of its domain of definition. Today this set is called the clugter set of a function at a point. Although the theory was first considered for analytic functions, it is applicable to more general functions, and much of the present-day research is largely topological. Early developments in the theory of cluster sets were mostly con- cerned with the behavior of an analytic function in the neighborhood of an isolated essential singularity or in a discontinuous set of singu— larities. The earliest result dealing with cluster sets was the theorem proved in a paper of Weierstrass (1) in 1876. It states that if 20 is an isolated point of a set E in the unit disk D and f(z) is meromorphic in,D - E, then the set of limit points of f at z0 is either a single Point or the entire Riemann sphere. In 1905 Painleve proved that this theorem is true for any 20 in a set of measure zero. If E is allowed to contain a continuum, then the cluster set of f itzonny be 3 pr has been done co simply connected which can theref The study 0 iatou paper (1) the unit disk. between the unit conformal mappin prudence between the danain whose function at the Since the 1 u item 2: c1 SLtsby Noshiro In this thesis We and present sane We organize three major clas and Class A Func Names by Chapter on orbit in for introduc last chapter use flholcmorphic I may of Class A functions in the at 20 may be a prOper subset of the Riemann sphere. A lot of research has been done concerning the boundary behavior of functions defined in a simply connected domain whose boundary contains more than one point and which can therefore be mapped conformally onto the Open unit disk. The study of cluster sets at a continuous boundary begins with the Fatou paper (1) of 1906 on the radial limits of functions analytic in the unit disk. Caratheodory (4) studied the boundary correspondence between the unit disk and an arbitrary simply connected domain under a conformal mapping. This led to the notion of a prime end, the corres- pondence between the points of the unit circle and the prime ends of the domain whose impressions are the cluster sets of the mapping function at the corresponding points. Since the 1930's, cluster sets have been widely studied. The books It}; Theory pf Cluster Set; by Collingwood and Lohwater (1) and Cluster §_e_t_s_ by Noshiro (1) contain most of the important results before 1960. In this thesis we assume the above material to be background information and present some of the more significant developments since then. We organize our material into three chapters which deal with the three major classes of functions: Arbitrary Functions, Normal Functions, and Class A Functions. We have selected results from The Theory gf_ Cluster4§g§g by Collingwood and Lohwater (1) for preliminary work in the chapter on arbitrary functions and results from Cluster Sets by Noshiro (1) for introductory material in the chapter on normal functions. The last chapter uses as background the MacLane paper (1), Asymptotic Values 91; Holomorphic Functions, since this paper is the foundation for the study of Class A functions, which consist of non-constant holomorphic functions in the open unit disk which approach limits on a dense subset of the unit circle included without p In the first disk without impos ulued. In spite uited patterns of is the result of Bugauihl's United patterns a cmplex-valued able number of be curves in D endin though this resul of the function 1 over the addition the to be analyt Seldel, h). This to other domains, approximately the The above td he sets. For e: point p if, for a: with are mutual cluster sets of nonempty. Then lltisfying the t fluently these pt 3 of the unit circle. All of the results from these references have been included without proof. In the first chapter we primarily consider functions in the unit disk without imposing any restrictions except that they be complex- valued. In spite of the lack of restrictions, these functions have li— mited patterns of behavior with respect to cluster sets. Much of this is the result of the particular topologies of the plane and the sphere. Bagemihl's Ambiguous Point Theorem is an outstanding example of the limited patterns of behavior mentioned above. The theorem says that if f is a complex-valued function defined in D, then there are at most a count- able number of boundary points g with the property that there exist two curves in D ending at g along which f has disjoint cluster sets. Even though this result is true in the plane, it does not apply if the domain of the function is the unit ball in three dimensions (Church, 1). More- over the addition of some mild restrictions, such as requiring the func- tion to be analytic, does not yield a stronger conclusion (Bagemihl and Seidel, 4). This theorem has been extended by the theory of prime ends to other domains, such as simply or multiply connected regions, with approximately the same result. The above theorem has found wide application in the study of clus- ter sets. For example, let f have the n-separated-arc property at a point p if, for any integer n> 1, there exist n arcs in D ending at p which are mutually disjoint except for p where the intersection of the Cluster sets of all n arcs is empty while that of any n- 1 of them is nonempty. Then if f is a homeomorphism of D onto itself, any point p satisfying the n-separated-arc property is an ambiguous point. Conse~ Cluetltly these points are at most countable. However, this does not hold in general as the: arc property at al in interestin relationship betwe angular cluster se angle at a point e set of a function end an angle have tween these two t there are numerou maglobal, not points e19 on the horocycle such th horocyclic cluste r is called : lection of arcs w ur(£,eie°) is th I‘uhich end at th the union of the 010 ~ eieol < u. the intersection continuous functi set is equal to 1 first category 01 chords . 4 in.general as there is a continuous function which has the 3-separated- arc property at all but at most a countable number of points (Piranian, l) . An interesting relationship between different cluster sets is the relationship between the angular and the horocyclic cluster sets. An angular cluster set is the cluster set of a function relative to an angle at a point eieesc while a horocyclic cluster set is the cluster set of a function relative to a region between two circles each intern— ally tangent to the unit circle at a common point. Since an angle is the region between two chords originating from eie, a horocyclic region and an angle have no points in common near eie. So a relationship be- tween these two types of cluster sets should not be expected. However, there are numerous relationships between these two kinds of cluster sets on a global, not local, basis. For example, for any function the set of points e19 on the unit circle for which there exists an angle and a horocycle such that the angular cluster set is not contained in the horocyclic cluster set is a set of measure zero and of first category. P is called a selector of arcs if it associates a nonempty col- lection of arcs with every point in C. The P-principal cluster set Ilr(f,e 0) is the intersection of the cluster sets of all the arcs in ieo 190 l‘which end at the point e . Letjn&?(f,e ,M) denote the closure of the union of the P-principal cluster sets for all points e19 for which leie - e190, < #. Then the boundary r-principal cluster set at eieo is the intersection of the Ilr(f,eieo,u)‘s for all positive n. For any continuous function f in D and all points in C, the r-principal cluster set is equal to the boundary reprincipal cluster set except for a set of first category on C if T is either the collection of all arcs or all chords. For continuou Jordan arcs in D e note between any between a point in topology is cells then the set (:(e1 in the ii-topology this the fact t is at most counts The cluster extensively. As possess properti of arbitrary fun queutly are no Any normal out a point 20 let limit a at 2‘ continuous curve lute than one p0 where f(z) must tent along a seq Analytic nr the special type closed under add at?) If'(z)| is satisfies the c collection of a 5 For continuous functions defined in D, the cluster sets along all .kndan arcs in D ending at a point on C form a topology where the dis- tance between any two closed sets is defined as the greatest distance between a point in one set and a nearest point in the other set. This topology is called the M-topology. If e19 is not an ambiguous point, then the set G(ele) consisting of its Jordan-arc cluster sets is compact in.the M-topology. So another consequence of the Ambiguous Point Theo— '6 rem is the fact that the set of points for which C(e1 ) is not compact is at most countable. The cluster sets of special classes of functions have been studied extensively. As might be expected the cluster sets of these functions possess preperties which are not necessarily true for the cluster sets of arbitrary functions. Some of the functions investigated most fre- quently are normal functions and Class A functions. Any normal meromorphic function f(z) in D which approaches a limit a at a point 20 in C along a Jordan curve lying in D also has the angu- lar limit a at zo. Moreover, if f(z) tends to a limit along a simple continuous curve z(t) for which lz(t)b4>l as t-a-l and its end contains more than one point, then it is a constant function. Another example where f(z) must be a constant function occurs when it approaches a cons- tant along a sequence of arcsier which convergeix>a boundary are in C. Analytic normal functions are not closed under addition although the special type of functions called uniformly normal functions are closed under addition. These functions satisfy the condition sup(fl.- 2 zel) lZl )lf'(zfl is finite. If, in addition, a uniformly normal function f satisfies the condition f(0)==0 then it is called a Bloch function. The collection of all Bloch functions form a Banach space. Each Bloch function, and thus it possesses angul Any normal ho which consists of that if there exis ”PEG, then ale bounded. If f is set of all clusn connect in the M-‘ function the set i In order to C, D has been com any bounded holom dense in M. No hounded holomorph mush“ % has t and f(mz) is Stri relation. E ach G Image of a One.“ Parts partition t analytic funct 1m {mum is norm < t othe set C Conn ii to 131131 functions no nal “Morph. t ‘ . nunal Gleam 6 function, and thus each uniformly normal function, has the property that it possesses angular limits on an uncountably dense subset of C. Any normal holomorphic function in D belongs to the class I , which consists of those holomorphic functions f in D having the preperty that if there exists a pair of arcs t1 and t2 along which f(z) +00 as z->peC, then along any path between t1 and t2 the function f(z) is un- bounded. If f is in class Ip, then the set G(eie) consisting of the set of all cluster sets of all Jordan arcs in D which end at p is compact in the M-topology. Consequently, for any normal holomorphic function the set G(eie) is compact in the M-topology. In order to study the behavior of normal meromorphic functions near C, D has been compactified into a Hausdorff space M in such a way that any bounded holomorphic function f has a continuous extension f and D is dense in M. Theo points m1,m2 EM are in the same Gleason part if for any bounded holomorphic function of modulus less than or equal to one, its extension f has the preperty that the difference in magnitude of f(ml) and f(mz) is strictly between 0 and 2. This determines an equivalence relation. Each Gleason part consists of either a single point or the image of a one-to-one analytic map of an open disk into M. The Gleason parts partition the boundary points of D in such a way that any bounded analytic function has a continuous extension onto the boundary of D. A function is normal in D if and only if it can be continued continuously to the set C consisting of the maximal ideal space M of H°° minus the tri- vial Gleason parts lying over the boundary of D. So, in this sense, normal functions are a generalization of bounded functions. If f is a normal meromorphic function, then it is so continuous that on every non- trivial Gleason part f is either meromorphic or identically infinite. Let f(Z) be a longs to Class A i has a path in D en belongs to Class E npath in D ending finite is dense or sists of all poin' Class L if and on level set interse theorem in the st Class A if and on function is in C1 In order to defined by replac al’f’roilriate defir and meromorphic r in Class Brn' Cl: mm“ 0f functir functions in 013; Atract ass donning 0(6) Sue Which is mafiled tersection of al nonempty, COMM uni)r if its end 7 Let f(z) be a holomorphic nonconstant function in D. Then f(z) be— longs to Class A.if and only if for each point in a dense set of C, it has a path in D ending at e19 along which it approaches a limit. f(z) belongs to Class B if and only if the set of points e116 for which it has a path in D ending at eie along which either f—>°° or IfI < a where a is finite is dense on C. For any constant k 2;0, the level set LS(h) con- sists of all points z in D for which lf(z)| = h. Then f(z) belongs to (Hess L if and only if the maximum.diameter of the components of each level set intersected with {2: Izl > r} ->0 as r +1. A very important theorem in the study of Class A functions states that a function is in Class A if and only if the function is in Class B if and only if the function is in Class L. In order to generalize Class A functions, Classes Am, Bm and Em are defined by replacing the word "holomorphic" with "meromorphic" in the appropriate definitions. Holomorphic normal functions are in Class A and meromorphic normal functions are in Class Bm. Class Am is contained in Class Bm. Class Lm is contained in Class Em. However, there are ex- amples of functions in Class Em that are not in Class Am and examples of functions in Class Em that are not contained in Class Em. A tract associated with a constant a is a collection of nonempty domains D(e) such that each D(e) is a component of the open set in D which is mapped by f into the open disk about a of radius 6 and the in- tersection of all of the D(e)'s is called the end of the tract. It is a rmmempty, connected closed subset of C. A tract is called global if and only if its and consists of the entire circumference C and for each arc Y contained in C there exists a sequence of arcs Yn contained in D(1/n) such that the Y 's approach Y. If f is in Class A and {Y } is a n m n sequence of dis joi Stiff“) - a) +0 n dewhich contai only asymptotic ve global tract if an to +1. If a is a fi' one onto a linear accessible. If f set of linearly a Class A func and multiplicatic Dean be written Class A. Further represented in ca holomorphic fUDCI Class A and a for functions in 01a: lemmas may be and Lohwater, an 8 sequence of disjoint simple arcs in D which tend to the arc Y on C and sup(f(z) — a) ->0 where a is a complex number, then f has a tract with end K which contains ‘Y. In addition for any interior point of K, the only asymptotic values come from the tract. If f is in A, then f has a global tract if and only if f is unbounded on every curve in D on which lzl -) 1. If a is a finite asymptotic value along an are that f maps one-to- one onto a linear segment, then this asymptotic value is called linearly accessible. If f is in Class A and omits some finite constant, then the set of linearly accessible points is dense on C. Class A functions are not closed under the operations of addition and multiplication. In fact every nonconstant, holomorphic function in D can be written as the sum or as the product cf pairs of functions in Class A. Furthermore, any nonconstant meromorphic function in D may be represented in each of the following ways: (i) the quotient of two holomorphic functions in Class A, (ii) the product of a function in Class A and a function in Class AIn 0 Class Lm, (iii) the sum of two functions in Class Am 0 Class Lm' Our bibliography consists primarily of the publications which con— tain the recent deve10pments in the theory of cluster sets. Older re- ferences may be found in the bibliographies of the books of Collingwood and Lohwater, and Noshiro. In this sect nitions which wil some of the major cluded in the boc We will cone [0! the unit circ 30‘“ 0f the chmter sets, as: nun Maud f §£clhzol of f talent WayS: (l) C(f,z 0 for wh that, D‘lz c (in) For 1- and d1 CHAPTER I QBITRARY FUNCTIONS INTRODUCTION In this section we will first introduce some of the important defi- rfitions which will be used throughout the paper. Then we will summarize some of the major results in the theory of cluster sets which are in- cluded in the book by E.P. Collingwood and A.J. Lohwater (1). We will consistently use the notation D for the open unit disk, C for the unit circle, and W for the Riemann sphere. Some of the concepts which we will use repeatedly include those of cluster sets, asymptotic values, and range of values. If 20 is any Point in D and f is an arbitrary function defined in D, then the cluster §££,C(f,zo) of f(z) at 20 is defined in one of the following two equi- valent ways: (i) C(f,zo) is the set of points a on the Riemann sphere W for which there exists a sequence {Zn} in D-{zo} such that, as n +>cn, limzn = Z0 and lim f(zn) = a where D-{zo} is D with 20 removed. (ii) For r >'0, C(f’zo) = DD; where Dr = f(drrl(D"lZo})) and dr is the disk |z«-zo| <'r. fit is any infini f(z) relative to G where Dr(G) = f(d] The range g_f_ such that there e: andz i‘ z , lim 0 0 diff!) at 2 con 0 0 exists 3 continue 1i“(0‘2 and o 5W1 W) to tie inC and the symi If Y = z(t: except for 'p = 2 My lff is an arbitrary Band terminatin Aset E on setof “Where ( to be of s m Complement of E Fot C any 9 . och point z E 10 If G is any infinite subset of D, then the cluster set CG(f,z ) of o f(z) relative to G is defined by CG(f,zO) = n Dr(G)C C(f,zo) where arm) = f(dr n (n - {zo})nc). The range 9_f_ values R(f,zo) is defined to be the set of values a such that there exists a sequence {Zn} in D such that as n —-—> co and z 7‘ z , lim 2 = z and f(z ) = a. The set of asymptotic values n o n o n -— "- A(f,zo) at 20 consists of those complex numbers a. for which there exists a continuous curve 2 = z(t), 0 < t < 1, SUCh that z(t) C D ‘{ZO}> lim z(t) =20 and lim f(z(t)) = a as t —> 1. We will. use the symbol A(f) to denote the union of all of the A(f,zo) 's for all zo's in C and the symbol R(f) to denote the union of all of the R(f,zo)'s. If Y = z(t), 0 S t S 1, is a simple continuous arc lying in D except for T = 2(1) 6 C, then ‘r is called a boundary are at T. limit; l: If f(z) is an arbitrary function defined in D and if g is an arbitrary point of C, then there exists a simple arc 'r, lying in D and terminating at I; , such that CY(f’C) = C(f,§). (Collingwood, 2) A set E on C is of first category if E is the union of a countable set of nowhere dense sets; a set which is not of first category is said to be of sicond category. A set E on C is called residual on C if the complement of E on C is of first category- For C any subset of B, a rotation Ge of G is obtained by mapping 19 each point z e G to the point ze my If the and if {as} is the com; is the poin of points e16 on C Let [3(1) be 5 is equal to {1}, E rotation about the p.58) proves that of C. In order to < 11 Theorem 2; If the real or complex function f(z) is continuous in D and if {G6} is the family of rotations of a continuum Go such that Gon C is the point 2 =1, then CGe(f,eie) = C(f,eie) on a residual set of points e16 on C. (Collingwood, 3) Let A(1) be an open connected subset of D such that A(l) (W C is equal to {I}, and let A(ele) denote the transform of A(l) under the rotation about the origin that sends 1 into e16. Dragosh (2, Lemma 1, p.58) proves that C . (f,eie) = C(f,eie) for a residual G8 subset A(e19) of C. In order to define boundary cluster sets, we use the notation C(f, 0 < 'e - 90' < n) = L) C(f,ele). (1) where the union is over 0 < '9 - 90' < n. Then the boundary cluster §S£ CB(f,eleo) may be expressed as o c (f,ei90) = n C(f, o < lo - e l< n). (2) Eb I!>0 o 16 The l2££;h§3§ and right-hand boundary cluster sets CB1(f,e ) and CBr(f,ele) are defined by (l) and (2) and the restrictions that 0 < 9 ‘ 90 < n and 0 < 90 - e < 0 respectively. Th ~JEEEE,§: If f(z) is a single-valued (real or complex) function in D, then 19 = is = f 19 Cr(f,e ) C31(f’e ) C( ,e ) 1 . except Perhaps for a countable set of points e 9 e C. (Collingwood, 4) The M'EEE points n such that 1n9n= 6 with at is defined in the right-hand clustei disk closed relati right of it. The is to the left of Corollary: If f( except perhaps f0 Lohwater) 1, Coro function defined Peri)! that there i The Poi] poi“ts. 12 The right-hand cluster set CR(f,eie) is defined to be the set of points or such that as n +u lim f(rneien) = or where lim rn= l and lim 9n = 9 with en 5 9n+1 g . . .. The left-hand cluster set CL(f,eie) is defined in the same way except that en 2¥en+l 2_.... Actually the right-hand cluster set is the cluster set CG(f,eie) where G is the semi- disk closed relative to D with diameter from -e19 to eie and to the right of it. The left-hand cluster set is defined in a similar way but is to the left of the diameter. Corollary: If f(z) is single-valued in D, then 9 6 i_ is= i CR(f,e ) - CL(f,e ) C(f,e ) except perhaps for a countable set of points e16 e C. (Collingwood and Iohwater, 1, Corollary, p.83) IQ§£EEE§;(Bagemihl Ambiguous-Point Theorem): If f(z) is a complex .9 . function defined in D, then the set of points e1 on C With the pro- i9 Perty that there exist two boundary arcs r1 and r2 at e such that . e c (f,ele) n c (f,ei ) = o r1 r2 is at most countable. (Bagemihl, 1) Tile pcnirrts eie defined in Theorem 4 are called ambiguous POints. RESULTS REL Researchers, proved many theore Let a be an a h'oi p and the uni called an extends: l-{p‘ such that 5-! 1mg: If f 0110 is an extend for f. (H. Mathe‘ Since Mathew recestly publishe SUCh that o tends tends to 1) such t and the method of Ellery Sequence of hints only in H i s taken over 31] 13 RESULTS RELATED TO BAGEMIHL'S AMBIGUOUS-POINT THEOREM Researchers, such as Bagemihl, H. Mathews and McMillan, have proved many theorems related to the Bagemihl Ambiguous-Point Theorem. Let a be an are lying in D - {p} except for one end point at p. The extended arc cluster set of f at p, ECa(f,p), is defined to be the set fl FEED-where the intersection is taken over all neighborhoods N of p and the union over all q on 301‘! for q 3‘ p. The point p is called an extended ambiguous p_oint for f if there exist arcs a and B in D - {p} such that ECa(f,p) and ECB(f,p) are disjoint. 1295312131: If f is an arbitrary function defined in D and if a point p on C is an extended ambiguous point for f, then p is an ambiguous point for f. (H. Mathews, 1, Theorem 1, p.138) Since Mathew's proof only holds when f is continuous, Stebbins (1) recently published the following proof. Let a be any are in D - {p} such that (1 tends to p. It is sufficient to find an are or' C D which tends to p such that Ca,(f,p) g ECa(f,p). By using points q 6 or n C and the method of Cross (1), we construct a ”wedge" Z in D such that every sequence of points {2k} in Z tends to p and {it-(2k); has limit points only in n mwhere q e anC for q 7‘ p and the intersection is taken over all neighborhoods 0f P- M: An arbitrary function from D into W can have at most a countable number of extended ambiguous points. (H. Mathews, 1, Theorem 2, p.139) This corollar guous Point Theore TM is a sin then we denote the and if there exist called an a_cc_ess_i_i AJordan arc Jordan curve whic! QU_toi G. A sequ flailif the £011 (1) No two common; (ii) qn sepa and the by d ; (iii) The die In chains Q = {( Values Of “a the cIOSSiltltjs q I 11 am the crosscuts q ‘ n is an equivalenc. Chams in G Ac was ”his on A that an 3 in each (in. If 14 This corollary follows immediately from Theorem 5 and the Ambi- guous Point Theorem. If G is a simply connected region in the extended complex plane, 16 e F(G) . . . . . i i , and if there ex1sts an arc 1n C With an end pornt at e 6, then e 9 is then we denote the set gf_boundary points of G by F(G). If e called an accessible point of F(G). A Jordan are which lies in G except for its two endpoints or a Jordan curve which lies in G except for one point is called a cross- 92; of G. A sequence ql’ q2,..., qn,... of crosscuts of G is called a 22213 if the following conditions are satisfied: (i) No two of them have any point, including their endpoints, in common; (ii) qn separates G into two domains, one of which contains qn-l and the other qn+1. The domain containing qn+1 is denoted by dn; (iii) The diameter of qn tends to zero as n tends to infinity. TWO Chains Q = {qn} and Q' = {qn'} in G are eguivalent if, for all values of n, the domain dn contains all but a finite number of the crosscuts qn' and the domain dn' contains all but a finite number of the crosscuts qn. The class of all chains equivalent to a given chain is an equivalence class. A pgim§.§gd of G is an equivalence class of Chains in G. A SEEKE 11 in G at Egg 23132 end P means a simple continuous Curve 2 = z(t), 0 S t :S 1, such that z(t) E G and every sequence of Points on A.that approaches F(G) also converges to P in the sense that all but a finite number of the members of the sequence are contained in each dn' If e19 e F(G) and there exist distinct prime ends P1,P2€(3 and curves r and s arcs at cm, then accessible point : If A is an a: G), then the clus Chiheie) [or CA(' curves r and s at .anbiggous prime g Mill 9 A nec legion G must sat more than countab that at most Coun Slble frm G. (B PM: SuPPOse t from is more th conformal manner F(G) and c under _ hints of C. Thu Assnne that ha one't0‘0ne c @18qu Mint e ah) 5 3“ (W) in iESPOHds to em U P . out Theorem tha hints 15 and curves r and s at P1 and P2 respectively such that r and s are also arcs at e19, then eie is a multiply accessible poi_n_t_ of F(G). If an accessible point is not multiply accessible, it is simply accessible. If A is an are at a point eieeF(G) (or a curve at a prime end P of G), then the cluster set of f at e19 (or at P) on A will be denoted by CA(f,eie) [or CA(f,P)]. If _P is a prime end of G and there exist two curves r and s at P such that Cr(f,P)nCS(f,P) = ¢, then P is called an ambiguous prime end of f. Theorem 9: A necessary and sufficient condition that a simply connected I region G must satisfy, in order that every function defined in G have no more than countably many ambiguous points from different prime ends, is that at most countably many accessible points of F(G) be multiply acces- sible from G. (Bagemihl, 5, Theorem 8, p.203) m: Suppose that the set M of all points of F(G) multiply accessible from G is more than countably many. Let w= f(z) map G in a one-to-one conformal manner onto D. This mapping induces a correspondence between F(G) and C under which every point of M corresponds to at least two pOints of C. Thus f has more than countably many ambiguous points Assume that F(G) contains at least two points. Let z=¢(w) map D in a one-to-one conformal manner onto G. If a function g(z) in G has an ambiguous point e16 that is simply accessible from G, then the function MW) ‘3 8(¢(W)) in D has an ambiguous point at the point w on C that cor- reSponds to e19 under the mapping ¢- It now follows from the Ambiguous Point Theorem that g(z) has no more than countably many ambiguous points. W1: Let f( gionGwith at lea terably many ambit PM: Let 2 = W Caratheodory's Th1 one-to-one corres Gsuch that, if P the preimage of A that corresponds many ambiguous pr have more than er M i: SUppc set WIS) 0f poj t i hat came 9) C C. Then the set Csiftelems at s The thEorem [l e cut T at e19 Wi Huh} Where n Euclidean diStan ii If f is any snenetric Spa: 8 11th that for Qt 16 Theorem 1: Let f(z) be an arbitrary function in a simply connected re- gion G with at least two boundary points. Then f has at most enu- merably many ambiguous prime ends. (Bagemihl, 5, Theorem 9, p.203) Proof: Let z = ¢(w) be a one-to-one conformal mapping of D onto C. By Caratheodory's Theorem (Caratheodory, 3 and 4) this mapping induces a one-to-one correspondence between the points of C and the prime ends of G such that, if P is a prime end of G and A is a curve at P, then P the preimage of AP under the mapping is an arc OT at the point T of C that corresponds to the prime end P. If f has more than enumerably ! many ambiguous prime ends, then the function h(w) 5 f(0(W)) in D would have more than enumerably many ambiguous points, which is impossible. 111m 8.: Suppose f is continuous, S is a closed subset of W and the . ie . 19 h set 305,3) of {30111128 e for which there ex1sts an are d at e suc that 0603,6319) C S is uncountably dense on an arbitrary closed arc h on 19 G. Then the set B*(f,S) of points e19 such that for any arc 0 at e Cg(faeie)n3 7‘ 4' is residual on k. (McMillan, 2, Theorem 5, p.188) The theorem is proved by showing that B*(f,S)flInterior(}\) rela- tive to C = 9{eie in the interior of A such that there exists a cross- Cut T at e16 with diameter less than l/n such that f(T) C {wz 9(w,S)< l/n}} where n is a positive integer and 9(w,S) denotes the Euclidean distance between w and S. If f is any function that is defined in D and takes its values in some metric Space, then a boundary function for f is a function 4’ on C SUCh that for every x e C there exists a simple are A having one endpoint at x for list f(z) = ¢(x)- functions. (3383' M: By the Am higuous points. values. Therefor In 1965 Kan: functions definec lll terms of honor “Ming: If : boundary functiou i0 is continuous N. (Kaczyns' Let 3* be a of all points on them“ is prove (i) aec(f‘ iii) if u 1 U=U (Snnn l7 endpoint at x for which A - {x} C D and as 2 approaches x along A lim f(z) = ¢(x). R Theorem 3: Every function f defined in D has at most 2 0 boundary functions. (Bagemihl and Piranian, 1, Theorem 1, p.201) Proof: By the Ambiguous Point Theorem, f has at most countably many am- biguous points. At each ambiguous point f has at most 2N0 asymptotic 80 values. Therefore, f has at most (230) =2x° boundary functions. RESULTS ON BOUNDARY FUNCTIONS In 1965 Kaczynski published a paper on boundary functions for functions defined in D. It includes descriptions of boundary functions in terms of honorary Baire class functions. Wig: If f is a homeomorphism of D onto itself and ¢ is a boundary function for f, then there exists a countable set N such that (be is continuous where (b is the restriction of the boundary function to o C ‘ N. (Kaczynski, 1, Theorem 1, p.590) Let 8* be a base of open sets in R2 and let acc(E) denote the set Of all points on C which are accessible by arcs in E. Then the above theorem is proved by showing that for any 3 e 3* (i) acc(f‘lums» = acc(f'lmn 5)) n f_1(D - s) u (c- f-1(D — 3)) (ii) if U is any open set which can be expressed in the form — - -1 U = US where S E 8* and Sn 9 U, then ¢01(U) = Uacc(f ( n n (S 0D)) - N where N consists of all of the ambiguous points n accessib ppgl: Let f be finite-valued bout t< t. Then (A) there e: where a of Open (3) there e (Kaczl’nski, 1, Le M: LEt n be 2 =1‘1/n},, ”lists an arc Y M) £1!)sz [XE +0ill}. F01“ a f dill Sllch that Y [19¢th 36!: S of the Open set iii T be the set olE . ll 18 accessible by arcs in f-1(DrlU). lemma l: Let f be a continuous real-valued function in D and k be a finite-valued boundary function for f. Let r and t be real numbers with r‘< t. Then (A) there exists a G6 set G and a countable set N such that -l -l l ([r,+°°))2G2>t ([t,+°°)) -N where a G6 set is the intersection of a countable number of open sets and (B) there exists a G5 set H and a countable set M such that i'1((-oo,t]) 2 H2 i'1<(-oo,r]) - M. (Kaczynski, 1, Lemma 3, p.592) m: Let n be any positive integer. Let 6 - (t-r)/2, Cn = {zeR : 2 '2' = 1 ' l/n}, An: iZER: l> '2' > l-l/n}, En: (xeC: there exists an arc y at x having one endpoint on Cn with y - [x] E f'1((-«5r ))], and K.= {xesCz there exists an arc y at x with y - {x} S f-1((t"€a +“0)}- For a fixed n and any point x in K'we can find a simple arc Yx at x such that Yx - {x} 9 Ann f"1([t-e,+°°)). Then Yx - {x} is a con- nected set. So yx-{x} must be contained entirely within one component of the open set Anrlf-1((t-e;hw)). Let 0x denote this component and lfit T'be the set of all points of K which are two-sided limit points ofli. n llevaflt to Sh .Ity set. SUPPOS lxlloy- We choos ively. Then we jo arc in 0x- Simfla syasubarc of Yy' vith endpoints at its is not a sin}: tho having endPOi ' crosscut of D. LE sand y. Accordir couponents V1 and ively. Because Cl taiued entirely w: some thatC is n hint of in’ L1 mu Suppose w is an e joining to to some cannot have a poi hell and f.1((-o aconnected set 11 My Consequen illitadiction bec qu“my if X y e T18 countab 19 We want to show that if x,y e T and x 5‘ y, then 0x 0 0y is the empty set. Suppose on the contrary there exists an element 2 in Oxrl 0y' We choose points x' and y' in Yx - {x} and Yy - {y} respect- ively. Then we join x to x' by a subarc of Yx and join x' to z by an arc in Ox' Similarly we join 2 to y' by an arc in 0y and join y' to y by a subarc of Yy. Putting these arcs together, we obtain an are (I with endpoints at x and y such that a - {x,y} S An 0 f-1((t-e,+°°)). If a is not a simple arc, we replace it by a simple arc (1' contained in (1 having endpoints at x and y and rename the simple arc CL. on is a crosscut of D. Let L1 and L2 be the two open arcs of C determined by x and y. According to Newman (1, Theorem 11.8, p.119), D - (I has two components V1 and V2 whose boundaries are L1 U or and L2 U (I respect- ively. Because Cu is connected and does not intersect or, it is con- tained entirely within one component of D - or. By syrmnetry we may assume that Crl is contained in V2. Since x is a two-sided limit point of En, L1 must contain a point of En and hence a point 0f.En' SuPPose w is an element of L 0 En' There exists a simple are 6 l jOIning W to some point on on with B - {w} E f-1((-°°,r)). But 5 - [W] cannot have a point in common with or because (1 - {x,y} E f-1((t-e, +°°)) and f'1((-oo, r)) n f"1((t-e,+oo)) = o. Thus CnU (B - {w}) is a connected set not meeting or while meeting V2, and so is contained in V2. Consequently w is in the boundary of V2. However, this is a contradiction because w 6 L1 and the boundary of V2 is L2 U a. Conse- quently if x,y e T and x 7‘ y, then Ox 0 0y = o. T is countable since any family of disjoint nonempty open sets is countable. A180 t limit points of it ThenKllln = [Kn intersection of K is countable. Let Sinceh.1((-oo,r)) the ln's, C - [1. equal to l.1( [13+ 1 ii ([ts+°°)) ; so contains h.1( [t,+ (3) follows Lat S and T m if and 0:11 (1) domain (ii) range f (iii) there e Ping s lfunction g is C (1") domain (it) range E (ill) there ( able SE is eque hay functiol Q . ill If it 18 cm [is of 3&2 C v \ . . ...._.M_......o~——_. _ 20 countable. Also the set S of all points of E; which are not two-sided limit points of E; is countable. Again let n be any positive integer. Then ann = [Kns] u[rto ([t,+ )) “Clio ([rn,+ )y, And consequently for every rational t, h;1([t,+°°)) is a G5 set. If u is any real number, choose a strictly increasing sequence of rational numbers tn converging to u. Then l;1([ue+°°)) is equal to the . .. -1 Intersection over n of Aol([tn,+°°) ). Thus he ([Us+°°)) 13 a G5 set. . - -l Slmflarly A01(('°°,u]) is a GIS set for each real u. Therefore lo ((U,'*:°°)) is the intersection of an F0 set with C - N where an F, set is any set Which is the union of a countable number of closed sets. By a theorem of Hausdorff (1, l1 on C such that lIll(“,+“°)) is a class 1(c,n). Si Baire class 2(C,R Caroling: Let f b product of the 1 function for f, t 1, Corollary, p.5 PM: We expre mi“) and A = sols of honora Baire class 1(C, Setting g = (81,: that g agrees wi liiori=1,,,, fly Soppos M“ 0f R3, an ““0“ functio “id for all x e is“ 4. p.596: M: Let 3 tin“. If tl I 22 of Hausdorff (1, p.309), K0 can be extended to a real-valued function ll on C such that for every real number u, h;1([u,+°°)) is a G¢15 set and -1 . hl ((u,+°°)) is an F6 set. By Hausdorff (1, Theorem IX) K1 is of Baire class 1(C,R). Since h(x) = h1(x) except for x e N, A is of honorary Baire class 2(C,R) . Corollary: Let fbe continuous. If f : D —>RN where RN is the product of the reals with itself N times and h:C —-)RN is a boundary function for f, then X is of honorary Baire class 2(C,RN). (Kaczynski, 1, Corollary, p. 595) P_ro_of_: We express f and h in terms of their components: f=< f1,f2, ...,fN> and A = (h1,}\2,...,}\N> . hi is a boundary function for f1 and so is of honorary Baire class 2(C,R). Now we choose a function gi of Baire class 1(C,R) that agrees with hi except on a countable set Mi' Setting g = we see that g is of Baire class 1(C,RN) and that g agrees with A except on the countable set which is the union of . » N M1 for 1 = 1,”,N. Hence h is of honorary Baire class 2(C,R ). . . 3 . Latina; 2.: SuPpose g is a continuous function mapping C into R , q 18 a point 0f R3, and 6 is a positive real number. Then there exists a con- tinuous function g*:C -->R3 such that q does not lie in the range of g* and for all x e C, [g(x) - q] 2 f implies g(x) = g*(X)- (Kaczynski, 1’ Lama 4, p. 596) 23-92:: Let S be the set of points y in R3 for which ly - ql is smaller than G. If the image of C by g is contained in S, let 3*‘0 HR be nycontinuous ft pteimge of S, g.1 upreosed in the on for which 8k - Since g'1({q}) is ofinite number 0 points eia'k and e each k, a continul ink ib“ ‘zie ), sk(e fine 8*(x 3*( Tons 3*:0 —> R3 Thoma: If f Sphere ll and h is Baire class 2(C,h PM: Since W 1 "M is of hon ‘1“31(C,R3) wh “0) ' W is com 1”” in the m thug“ fUnc t 10 " ' coutinuouE range of 8* am It 23 any continuous function whose range does not include q. Otherwise the preimage of S, g-1(S), is a proper Open subset of C. Hence it can be expressed in the form g-1(S) = U Ik where Ik is the set of elements etszr which ak.< t.< bk and k # 1 implies that 1k and 11 are disjoint. Since g-1({q}) is a closed compact subset of g-1(S), it is covered by a finite number of the Ik's, say the union of I I ""In' The end- 1 3 2 3 iak ibk -l and e of Ik are not in g ({q}). 80 there exists, for points e ia each k, a continuous function gk frOm Ik into R3 such that gk(e 1‘) ib =18(e ak), gk(e1b k) = g(e k) and q is not in the range of gk. We de- fine g*(x) g(x) if x e C-(Il u I U...u I ) 2 n g*(x) gk(x) if x e 1 , k = 1,2,...,n. k Thus g*:C —)R3 as required. mg: If f is a continuous function mapping D into the Riemann sPhere W and h is a boundary function for f, then A is of honorary Baire class 2(C,W). (Kaczynski, 1, Theorem 3, 13-596) 3.12%: Since W is a subset of R3, the Corollary of Theorem 11 shows that h is of honorary Baire class 2(C,R3). Let g be a function of Baire Class 1(C,R3) which differs from A only on a countable set N. Then g(C) ‘ W is countable. Thus there exists a point q inside of W which is not in the range of g. Let gn be an element of a sequence of con- tinuous functions converging to g. By Lemma 2 there existS, for eaCh D, a continuous function gzzc —-)R3 such that q does not lie in the range of g: and, for all xeC, 'gn(x) - q 2 l/n implies gn(x) = gflx). M8500 approo He now want tleonique ray wi point of intersec l3- {4* onto W a fix i ll. P(g§(x +l(g(x)) as n-l mg: If t Baire function, t Ill is of Baire Baire class a. M: Let). be function for i. More than com ilereli is less ' Mm (1, Theorem Equal to the max Frequently M‘ For exam mad“? that: this “action m not. In later 24 Then g:(x) approaches g. We now want to define a function P. If a e R3 - {q}, let 1 be the unique ray with endpoint at q that passes through a and P(a) be the point of intersection of 1 with W. P is a continuous mapping of R3 - {q} onto W and P fixes every point of W. Therefore, P(g(x))==h(x) if x é N. P(g:(x)) is a continuous function from C into W and P(g:(x)) "* P(g(x)) as n—>oo. Theorem 12‘ If the function f has a boundary function A that is a Baire function, then every boundary function for f is a Baire function. If A is of Baire class a Z 3, then every boundary function for f is of Baire class a. (Bagemihl and Piranian, 1, Theorem 3, p.202) EEQQE: Let K be of class a and suppose that X1 is another boundary function for f. By the Ambiguous Point Theorem, Al differs from h at no more than countably many points; therefore, Al is of Baire class 5 Where 5 is less than or equal to the maximum of 2 and a according to Hahn (1, Theorem VII, p.352). By a similar argument a is less than or equal to the maximum 0f 2 and B. SOME SPECIAL TYPES OF CLUSTER SETS Frequently mathematicians have investigated special cluster sets of 3. For example, in our introduction of this chapter we mentioned boundary cluster sets and right-hand and left-hand cluster sets. In Hus section we will consider another type: the outer angular cluster set, In later sections we will consider some others. f A Stolz angle is a domain bounded by an arc of C and two chords 0 E the unit circle at one; ESE CAM sets C (f,ele) whl A lime-Ll: Let f b aandb are two f (ii/2. For each a 1")C Sn and (f,eie) is not contained in Sn} . We want to CA(e.j) Show that for each pair of positive integers j and n and each real num- ber r in the interval 0 < r < l, E(r,j,n) is of linear measure zero. 30 suppose that there exists a triple r,j,n such that E(r,j,n) is not 0f linear measure zero. Since E(r,j,n) is measurable and not of measure zero, there exists a subset E* of E(r,j,n) such that E* is closed, has n0 isolated points and has positive measure. Let G be the set [2: lzlér} U {A(9) :eie €E* ],where A(9) 19 as defined above . By an argument of Noshiro (1, p.71), the boundary of G is a rectifiable Jordan curve. 80 there exists a subset E. 0f 13* such that E' has positive measure and the boundary of G has a tangent at each point of E' and this tangent is the tangent to C at this point. For any 9‘ is contained in G h fre, C . ere 0 A“ ,J finition of E(r, j ZEIO. Su ose C PP No such that CA(G:j) sets is compact, ch(9,j)(f’eie> is let F(a,b) denote nonbers r betweet Since F(a,b) is I is of linear mean: 8on to cA(f,e1‘ W13 Let 26 point. For any point e19 e E' and some t > O, A(9,j)n {2: lz-eiel r, f(z) 6 Sn' Therefore, CA(9 j)(f,eie) is contained in Sn which contradicts the de- finition of E(r,j,n) . Consequently E(r,j,n) must have linear measure zero. S ie i9 . . uppose CA(9)(f’e ) 9‘ CA(f,e ). Then there must ex1st some J 19 19 . h that C f, C f, . S ach of these cluster suc A(9:j)( e ) 7‘ A(6)( e ) ince e sets is compact, there exists an integer n such that GAG) C 8n and C . (f,eie) is not contained in S . So for some r, e16 E E(r,j,n). A(99]) n Let F(a,b) denote the union of all of the E(r,j,n) 's over all rational numbers r between 0 and l and all pairs of positive integers n and 3'. Since F(a,b) is the countable union of sets of linear measure zero, it o i ' is of linear measure zero. If e19 e C - F(a,b), then CA )(f,e 6) 18 (9 equal to CA(f,eie). ill—993% ii: Let f be an arbitrary complex-valued function in D. Then there exists a subset F of C, where F is a set of linear measure zero, such that for each point e16 e C - F and each Stolz angle A with vertex at eie, CA(f,eie) = CA(f,eie). (Lappan,8, Theorem 1, p.1060; Brelot and Doob, Theorem 7, p.409) 2339.93 Let the elements of two sequences of rational numbers, denoted by an and bn respectively, satisfy the conditions -11/2 < an < bn < 11/2 and for each pair of real numbers c and d satisfying the condition ‘n/Z < C < d < n/Z there exists an integer n such that c < an < bn < d. Let F: 8 F(a ,b ). If e19 E C - F and if A is any Stolz angle with n=1 n n Vertex at 619 then there exists a positive integer n such that , Al(e) : 4| sodh'(e) is cont. ie : aodsoCA(f,e ) sets of linear me In the next satisfies the con has positive capa “(3) be a non-neg sets in the plane aconnected comp] the Property he now define the and the quantity hen the capacit capacity of any P1062). Let U 27 A'(e) = {z e Dzan < arg[l - (z/eie)] < bn} and A'(e) is contained in A. Since e19 of F(an,bn), = CA(f,ei9) C A'(o) and so CA(f,e16) = CA(f,ele) . Furthermore, F is the countable union of sets of linear measure zero. So F is also of linear measure zero. In the next paragraph we will give an example of a function which satisfies the conditions of Theorem 14 such that F is uncountable and has positive capacity. First we will explain the term capacity. Let #(E) be a non-negative additive set function defined on all the Borel sets in the plane. Let F be a closed bounded set in the plane having a connected complement G and M* be the set of all set functions It with the prOperty fduU’) = 1. {eF We now define the function 11(2) = f 103(1/lz-tl ) deem. HF and the quantity V =inf (su u(Z))o F u¢M* 2‘8 -V F Then the capacity of the set F is defined to be cap F = e , and the CsilPélCity of any Borel set E is cap E = sup (cap F). FCE We now give the following example which is found in (Lappan, 8: 9°1062). Let U be the upper half plane, P be the Cantor middle third set on the closed of open intervals let Tn be the tri hhaving In as it function f in U t If F is the subse then P C F; there In the stud: most important [I ahint e19 e C the hint eie. m“ 1) is t sidered to b e pa offateie 1ft 19 i . rim W analogously. I Suppose 0 . 28 set on the closed interval [0,1], and In be an element in the collection cflfopen intervals which are complementary to P in (0,1). For each n let Tn be the triangular region bounded by the equilateral triangle in 00 llhaving I; as its base. Let T = UlTn and V = U - T. We define the n: function f in U to be as follows: f(z) f(z) 0 for z E V, l for z E T. If F is the subset of C of linear measure zero mentioned in Theorem 14, then P<: F; therefore, F is uncountable and has positive capacity. HOROCYCLES In the study of cluster sets of special subsets of B, one of the nmst important types of subsets has been the horocycle. A horocycle at a point e16 E C is defined to be a circle internally tangent to C at the point eie. The horocycle is denoted by hr(eie) or just hr where r (0 < r'< 1) is the radius of the horocycle. The point eie is not con- sidered to be part of hr' A point w e W is a horocyclic cluster value of f at e19 if there exists a sequence with elements zn lying between two horOCYCles at e16 such that limzn = e19 and lim f(zn) = W- Given a horocycle hr at a point e19 E C, the region interior to hr is denoted by 91:. The half of hr lying to the right of the radius at i + i e 6 as‘viewed from.the origin is denoted by hr(e 0) and is called the £131E W at e16 with radius r. The left horocycle is defined analogously. In addition (I: and 0; denote the right and left half reSPectively of {Ir . ‘ t the Suppose 0'< r1 < r21< 1 and r3 (0'< r3 < l) is so large tha circle I ll = r33 hemm defined to be + rlarz’r here the bar der respect to the pl noted by Hrl r2 ‘ a " without specifyi‘ specify r1,r2,1‘3 he now wish cycles. The gig (when) = U C te‘r g of f at the intersection 29 circle '2' = r3 intersects both of the horocycles h r 16 and h . Then 1 r2 . . + the right horocyclic angle Hr at e with radii r1, r2 and r3 is 1 ’ r2 ’ 1'3 defined to be H+ ‘ comp 52+ n (2+ n{ I I > } ‘ z : z _ r , r13r2’r3 r1 r2 3 where the bar denotes closure and "comp" denotes complement, both with respect to the plane. The corresponding left horocyclic angle is de- noted by H;1,r2,r3° Hr1,r2,r3 denotes a horocyclic angle at e16 without specifying whether it is right or left. If we do not wish to specify r1,r2,r3, then the notation is simplified to H. We now wish to define special types of cluster sets for horo- cycles. The right outer horocyclic angular cluster set of f at ei6 is CU+(f,eie) = U CH+(f,eie), and the right inner horocyclic angular clus- Eerie; of f at e16 is CI+(f,eie) = n CHI-(Leia), where the union and the intersection are taken over H+ which ranges over all right horo- cyclic angles at e19. CU_(f,eie) and CI_(f,eie) are defined analo- gously. The Mar horocyclic angular cluster set of f at ei0 is defined to be CU = Cud— U CU" and the inner horocyclic angular cluster set of f is defined to be C = CI+ n CI" The right principal horocyclic clus- I Efil §_e_£ of f at e16 is defined to be I}: = n Ch+ while the left principal r WM; §§£ is defined by changing the + signs to - signs. The W hgrocyclic cluster set is the intersection of the right and the left principal horocyclic cluster sets and is similar to the W M cluster set which is defined as the intersection of Cx(faele) over X and denoted by flx(f,eie). Cx is the cluster set of f i . at e 9 on the chord X. The inner horocyclic angular cluster set 18 . i Simllar to the inner angular cluster set.CB(f,e 0) WhiCh is the intersection of i Finally we r 216 6C is caller cyclic Fatou valn the single w. A of t if C14, = W, , + provnded U (f ,e w right horocyclic right horocyclic + lef) respective no and M;( f) 0f horocyclic Fa Meier points of as follows: e19 In= 1:, 0 1;); e u, that is Mw is denote respectiu hints, ThESep in their Mules E uniform}, as 2 a for eyery angle C . up ~ Currie; 30 intersection of CA(f,eie) over all Stolz angles at eie. Finally we wish to define special types of points on C. Any point e19 e C is called a right horocflzlic Fatou point of f with right horo- cyclic Fatou value w e W whenever CU+ is equal to the set consisting of the single w. A point e5Le is called a right horocyclic Plessner point 9 is called a right horocyclic Meier point of f of f if CI+ = W, and e1 . + 19 _ ie . . prov1ded Hw(f,e ) - C(f,e ) is prOperly contained in W. The sets of right horocyclic Fatou points, right horocyclic Plessner points and , + + right horocyclic Meier points of f are denoted by Fw(f)’ Iw(f) and + _ M (f) respectively. The corresponding left horocyclic sets Fw(f), w 1;,(13) and M‘;(f) are defined in an analogous manner. Finally the sets of horocyclic Fatou points, horocyclic Plessner points and horocyclic Meier points of f are denoted by Fw’ Iw and MW respectively and defined as follows: e19 e Fw if CU is a singleton; el6 6 1W if CI = W, that is, = + - i9 . _ 19 . 1 d . I I n I ; e e M If H - C(f,e ) which 18 properly conta ne in W V W W W + - W, that is Mw is the intersection of MW and Mw' F(f), I(f) and M(f) denote respectively the sets of Fatou points, Plessner points and Meier points. These points are quite similar to those including "horocyclic" in their names since e19 e F(f) if CA is a singleton and lim f(z) exists 1 .. uniformly as 2 approaches e19 in any Stolz angle; e 9 6 H15} if CA " W for every angle A; e16 e M(f) if for any chord P(4’) of C P883318 1 through 619 and making an angle 4> with the radius to e 9, 47/2 < ¢> < "/2, C90b) = C(f,eie) which is properly contained in W. m it: Let f(z) be an arbitrary function from D into W. Then CB(f’eie) = CI(f,eie) = C(f,eie) for a residual G5 subset of C. (Dragosh, 2, Lemma 2, p.60) counts C (reo reside there cote; o—a =7‘ (D o / are ”a P1 [2 (D H- FD 31 10 Proof: For any e e C, let An r(e16) be the Stolz angle at eie with aper- , ture TT/2n, where the bisector of An r(e16) at e19 makes a rational angle r, -Tr/2< r sin(lrl +TT/2m-1) , then let A j—-A (eiefll a . Let n,r,m- n,r m E(eie) be the countable collection of all An r Inat 810 and Zw(ele) be the countable collection of Hr1 r2 r at e19 with rational radii ri. 3 3 3 i9 19 _ 19 . For each Ae}:(e ), CA(f,e )— C(f,e ) for a re31dual G6 subset of C (remark after Theorem 2). The intersection of countably many of these residual G6 subsets is again a residual G6 subset E1 of C such that C(f,eie)= fl . CA(f,eie)=CB(f,eie) for eleeEl. AEZ(ele) Also '0 i9 i9 i6 C(f,el )= fl - C (f,e )=C (f,e ) for e 6E H62 (e19) H I 2 w where E2 is another G6 subset and ElflE2 is the required subset of C. 31, 32 E C are topologically equivalent if 81-82 and 82-31 are of first category. + - Theorem 15: Let f : D +W. Then the sets I(f), IW(f), Iw(f), and IW(W) \‘ are topologically equivalent. (Bagemihl, 3, Theorem 4, P013) - - .6 06 ie — 2.19%: Since CI(f,e19)= CI+(f,ele) fl CI_(f,e1 ), e1 e C, CB(f,e )- CI+(f, i 19 e 9) = CI-(f,eie) = C(f,eie) for a residual set of points e on C. Th If f is an arbitrary function from D into W, then the sets Jamie: F(f): 1:“), F'(f) and F (f) are topologically equivalent- (Bagemihl, w w 3: Remark 3, 13.16) I23 32 Proof: By definition we have the following conditions: CB S CA, 01+ _ CCU...,C I' .. CU- and CI E CU for any point on C. Consequently Lemma 4 implies that CA = Cm. = CU' = CU for a residual set of points on C. In contrast to Theorems 15 and 16, the sets M(f) and Mw(f) are not ne- cessarily tepologically equivalent. (Dragosh, 2, Remark 3, p.61) For example, let S be a countable dense subset of C. We define f(z) in D as follows: f(0) = 0, f(z) = 1 for z e big-(e19) for e16 6 Sand f(z) = O for z e h;(eie) for e19 e C - 8. Since 8 and C - S are both dense on C, ”w is the element 0 for e19 E C - S and HW is the set with a single element 1 for e19 e 8. Thus M(f) = C, but Mw(f) = m2: If f(z) is an arbitrary function from D into W, then for any 19 is ie set L(e ) for which there exists a Stolz angle at e containing L(e ) CL is contained in CB except for a set on C which is of measure zero and of first category. (Dragosh, 2, Lemma 3, P~61) 2332.? Let E denote the set of points e16 E C for which CL is not con- . ' i . tained in CB. Then for each e19 e E, there exists a set L(e 6) lying inside of a Stolz angle at e19 for which CL is not contained in CA for some Stolz angle A at e19. So there exists a disk Qp on W SUCh that CL and Qp are not disjoint while CA and 6p are. Using the notation in Lemma 4, we can find a Stolz angle An r m p E 2(e 16) 31101" that “Am r :11) over all sub- and Qp are disjoint. So We can express E as ”En,r,m, i 9) Scripts where e16 5 En r m p if there exists at least one set L(e , 2 3 1Ying in a Stolz angle at e16 such that CL and Qp are 110': (118301!“- terse least cides fir 33 while f(An,r,m) and Qp are dis_101nt. Suppose there exists a set En r m p which has positive outer mea- 3 , 3 sure. Then f(A ) and Q are disjoint for e195 E ‘ . n,r:m P n,r:m:P I .= UA ‘ i9 ' f G n,r,m over p01ntS e e En,r,m,P’ then G is composed of finitely many open simply connected subsets G1, . . .,GN of D because C - E contains only a finite number of arcs with length exceeding n,r’m’p a fixed number between 0 and 21T. Privalow (l, p.220) has shown that each Gk’ for l ‘é k s N, has a rectifiable Jordan curve Jk as its boundary. Since En r m p is assumed to have positive outer measure, the in- , , 3 tersection of E and J must have positive exterior measure for at n,r,m’p k least one Jk' The tangent to Jk at almost every point of CO Jk coin- cides with the tangent to C. Consequently there exist points in k at which the tangent to Jk coincides with the tangent to C. At any such point each Stolz angle at that point has n,r,m,p belonging to C n J . , i . its terminal portion contained in Gk' So there eXist points e 6 in 19 19 ' t En,r,m,p such that CL 13 contained in f(Gk) for each set L(e ) a e which is contained in a Stolz angle at e19. Since f(An r m) and Qp are diSjoint for any point in E and G is the union over points in 3 $ 3 C O O ) and Q are disjoint. However, the definition P E f A f(G 0 n’r3m3p n,r’m, k of E says that for each point in E C n 9‘ ¢ for at n,r,m,p L Q? n,r,mip 1 least one set L(eie) lying in a Stolz angle at e 6 which is a contra- dietion. Therefore, each set E r m p has measure zero, and so E also n) ’ 3 has measure zero. By a similar argument it can be shown that each En,r,m,P is of fiI‘St category, and consequently E is of first category. .P_r_oo_f that zero 2 ant 31111 34 Theorem ll: Let f(z) be an arbitrary function from D into W and let K(f) denote the set of points e19 e C for which CA (f,ele) = (f,eie) 1 C , A2 for any pair of Stolz angles A1 and A2 at e16. Then K(f) is residual and of measure 217 on C. (Dragosh, 2, Theorem 2, p.63) Proof: At each point e16 E C - K(f), there exists a Stolz angle A such that CA is not contained in CB. By Lemma 5, C - K(f) is of measure zero and of first category. This theorem is a very important result as it generalizes Theorems 2 and 14. Theorem 18: Let f(z) be an arbitrary function from D into W and let ‘— Kw(f) denote the set of points e19 e C for which CHl = CH2 for any pair of horocyclic angles H1 and H2 at e19. Then KW is residual and of mea- sure ZTT on C. (Dragosh, 2, Theorem 3; 13.67) The theorem can be proved in a manner very similar to that of Lemma 5. Two sets S1 and 82 are called metrically equivalent if and only if measure (S1 - 82) = measure (82 - 81) = 0. W: If f(z) is an arbitrary function from D into W, then the + ' h ts 1+ 1' and I Sets Fw’ FW and FW are metrically equivalent and t e se w’ w W are metrically equivalent. (Dragosh, 7-, Corollary 1, P68) + .. m: Suppose eie belongs to at least one of the sets Fw’ FW and FW bUt not to all of them. Then there exists a pair of horocyclic angles 35 H and H at e16 such that C 5‘ C . By Theorem 18 the set of such 1 2 H1 H2 points e16 e C is of measure zero. So F w’ F; and FW are metrically + .. equivalent. The proof for Iw’ IW and IW is identical. F(f) and Fw(f) need not be metrically equivalent. Forexample (Dragosh; 2, Theorem 5, p.69), we are able to define the Blascke product a) 2n 2n 3(2) = I] ({n) + (Z3! n=1 (1 + {nz)2. where In = 1 - (n22n)-1 for any positive integer n which has zeros at the points z k = 1,2,...,2n and n.> o. _ f i(2k - 1)2'“n n,k _ ne ’ For each point f e C and each horocycle hr for O < r < l at the point, + - there exist sequences of these zeros lying interior to 9r and 91.. ThUS for each point in C, O 6 Co}? for 0 < r < l and similarly for CST . A r r Blaschke product has a Fatou value of modulus one at any point of C except for a set of measure zero. Let I be a Fatou point of B that has the Fatou value a with Ia' = 1. If i is a right horocyclic Fatou point of B, then 09+ is the set with the single element 0 for 0 < r < 1. r Since this contradicts the fact that CA is the set with the single ele- ment a for each Stolz angle A at f, the set of right horocyclic Fatou Points of B is of measure zero. By the corollary following Theorem 18, Fw(f) has measure zero. I(f) and Iw(f) also need not be metrically equivalent. Dra80Sh (1: Theorem, p.41) constructs a function f(z) holomorphic in D such that every point of C is a horocyclic Plessner point of f and almost every pOint of C is a Fatou point of f. 36 Lemma §_: If f(z) is an arbitrary function from D into W, then for any i set H*(e 6) for which there exists a disk Or at e19 containing H*(eie) i CH*(f,e 0) is contained in CI(f,eie) except for a set on C which is of measure zero and of first category. (Dragosh, 2, Lemma 6, p.67) Proof: Much of the proof of this lemma is analogous to the proof of Lemma 5. We replace Stolz angles by horocyclic angles and the region G . w . . . by a region G , which is defined as follows; let P be a perfect no- where dense subset of C and Hr r r (e19) be a fixed horocyclic angle, 1’ 2’ 3 w I then G is the union of all of the H 's for e16 in P. According r1,r2,r3 to Bagemihl (3, Lemma 1) GW is composed of finitely many simply.con- w w . . . nected subregions G , . .., k having as their respective boundaries the rectifiable Jordan curves JY, . . . , at almost every point e16 e C n J J So the tangent to J: for l 5115 k 8 5:8 coincides with the tangent to C. D We must now show that except for a set of measure zero contained in the set C 0J2, each horocyclic angle H at e19 has a terminal portion which lies in G: because the tangent to H at e19 also coincides with the tan- gent to C. In order to verify the last statement, we will first show that if P is a perfect nowhere dense subset of E0 , 1], then for almost every 13°int p E P for which a sequence of open intervals (an,bn) in [0 , 1] r P converges to 9, Ian - pl/(bn- an) tends to positive infinity. If E 18 any LebeBSue measureable set in R1 for which the upper and lower limits 0f the quotient meas (E 0(x - 6,x+ all 26 are equal, then their common value is called the metric density of E at x. According to Hobson (p.194), in our case the metric density exists 37 and is equal to l at almost every point p e P. Let p E P be a point with metric density equal to l and suppose that the sequence [ (an , bn) } converges to p from the right. Then meas(Pn (p,bn)) meas(Pn(p,an)) _ = lim n—-)oo lim 1 n _’ 0° meas(p,bn) meas(p,an) meas(Pn (p,bn)) and lim ——> 1. n —>oo (an - p)+ (bn - an) meas (Pn (p,bn)) Since Pn(p,bn) = Pn(p,an) , 1 im — a _ —+ l . n ———> 00 n P Also since meas(Pn (p,bn)), an - p and bn - an are each greater than zero, these conditions imply that lim[(bn - an)/(an - p)] approaches zero. Consequently (an - p)/(bn - an) ->+oo, and in general ‘an ‘13l/(bn - an) ->+°° . Now we will show that except for a set of measure zero contained in the set Cruz, each horocyclic angle H(eie) for e16 6 J: has a terminal portion which lies in 6:. By means of a bilinear transformation L(z), it is possible to map D onto the upper half plane and to prove this re- sult there. Let P be a nowhere dense set on the finite interval I on the real axis and {(an,bn)} C I - P. We now choose circles C1 : 2 2 2 2 (X ' an) + (y .. R)2 = R2 and CZ : (x - bn) + (y - r) = r where 0 0 let Ne“) denote the neighborhood consisting of ele- ments 2 such that 'z - g l < 6. Suppose we are given a set E in C and a point 3' on C. Let r(§',e) = r(§’,e,E) be the largest of the lengths of arcs contained in Ne “C and not intersecting E. Then for any a, O < as l, the set E is said to have porosity (a) at f if l-ijtl(r(i’,€))a/E >0 as e -—2 0. E is said to have porosity (a) on C if each point I in E has porosity (a). A set which is a countable sum of sets of porosity (a) is called a g—porosity (3) set. Yanagihara (1) has shown that E and EU are C sets and of UU V 60 d-porosity for some a. (Theorems 22 and 23) while Dolgenko (1) has shown that E is a G VV set and E is an F0 set (Theorems 21 and 24). 50 GV Theorem 21: If f(z) is an arbitrary function, not necessarily single- “* valued, then EVV is of G60 type and of d-porosity for some a. (Doléenko, 1, Theorem 1, p.3) m3 Let {an} denote a sequence consisting of all rational numbers between 47/2 and 1T/2, and let {a} be a sequence consisting of all closed circles in W - {00} having rational radii rn and centers at the P0111138 an with rational coordinates. If I E C, then Vp q denotes the Open angle of size a with vertex at f and with bisector forming an an- 819 “q with the interior normal to C at I. We define En p q to be the 3 ’ set of all points g' e C such that if z e D, P(Z,C) < 1/P, and for Z in VP”, the va 1 u e s of f(z) lie at adistancez rn from Dn where rn is the radius of DE: For m,n,s,k any positive integers, let Fn,m,k,s be the set of all points I e C for which the set 42 {f(z) :z is in Dfle,k and l/(3s) 5ru. m,k P,q Consequently we can find positive integers p and q such that if z is in Vp q and p(z,C) < l/p, then p(DJ,f(z)) > 4ru. Let n denote the index of the disk D_r:which has the same center as E and radius rn = 2ru. So P(¥,f(z)) > rn if z is in VP,q and P(z,C) < 1/p. Due to the choice of E, there exists a sequence of elements 2 in Vm,k which approaches f and a corresponding sequence of elements f(E) which approaches a point a E 3;. So for an infinite set of positive numbers, there exist points ‘2 in V , for l/(3s) < p(§,C) < l/s such that D—[: and {f(2)} are not disjoint. Thus 5* e 0L? F for all t and is therefore also in S=t n,m,k,s n,m,k' N . Then b the definition of ow suppose f e Fn,m,k and En,p,q y E — ' int. Since is in F , it is in n’p’q, CVp, and DD. are diSJO f n,m,k F for an infinite number of 5's. From the definition 0f n,m,k,s F . D— re not dis'oint and so C is n,m,k,s’ it follows that CVm,k and n a j Vm,k nOt Equal to CV P,q We will now show that E Suppose on the contrary W is a-porous . . , _ 'th there « and E Which is not 0 P0rous W1 eXists a point in Fn,m,k n,p,q 43 respect to G. Then the angle Vfi. close to its vertex is covered by a ,k union of angles V" for n in F and E . So by the definition P,q n,m,k n,p,q of E at points z in V ‘which are sufficiently close to the point, n:P:q m,k the values of f(z) are at a distance Zrn from.D;l Therefore CV k and m 3 D; are disjoint and the point is not in F k and E . Thus 3 3 3 3 F E is porous on C and E is a-porous. n,m,k n n,p,q W This theorem is closely related to the Collingwood Maximality Theorem (Collingwood, 3), which states that for an arbitrary single- valued function f(z) defined in D and any Stolz angle Atwith vertex at (9 93(f,() = C(f,() except for a set of first category. It is also re- lated to Theorem 14 which states that for an arbitrary single-valued function f(z) defined in D the outer angular cluster set CA = 9A ex- cept for a set of measure zero. Theorem.22: If f(z) is an arbitrary function, then EUU is of G60 type and of 0-porosity for some a, (Yanagihara, l, Theorem.l, p.424) The proof of this theorem is quite similar to that of Theorem 21. Theorem.23: If f(z) is an arbitrary function, then EUV is of Gfia type M—_ and of o-porosity' (0 ). (Yanagihara, 1, Theorem 2, p.425) Yanagihara (1, Theorem 4, p.426) has shown that there exists a bounded holomorphic function f(z) for which EUV is of measure 2n. For example, we pick an inscribed disk U(l)‘= {21 IZ ' Pl‘< 1 ’ Pi for 0 < P‘< 1. Then there exists a constant b such that an are A = iz- reie :9 = bfi/l-i:i is contained in U(l). In addition we choose tn 44 00 such that 0 < tn < 1, is strictly increasing to l and 2V1 - t 0 If 1‘ contains all arcs at 8160’ then the notation BH(f, 8160) is used. For any subset S contained in D, a point e16 e C is called almost L-accessible through S if for every Open set C with S 9G ED there exists an arc a e 1" such that a g G. This definition is abbreviated to e19 is almost accessible through S in the case that I‘ is the collection of all arcs at e19 which is a point of C. Let E be contained in C and 7 be a correspondence which associates with each point in E an arc 7(e19) in P(eie). Let S(7,E) denote the relative closure in D of the set 3(7aE) = U 7(eie) where the union is over all e19 in E. Then I‘ is a M W o_f_ eggs if for every set E of second category in C and every arc 7, there exists a subarc AEC such that E is dense in A and every point of A is almost F-accessible through S(7,E) . If I‘ is a selector of arcs, then a new selector of arcs 1"* called the completion 91:; I; is defined by {a : a QB e F(eiefl. Finally I‘ is called an gfimissible selector 2t. 1222 if 1"* is a smooth selector of arcs. The theorems which we prove in this section will lead to the maJor ' D result stated in Theorem 29 that if f is a continuous function in , 46 then H(f,ele) = BH(f,ele) and Fk(f,ele) = BIIx (f,ele) where X denotes the collection of all chords at e19 except fora set of first category. Theorem 22: If f is an arbitrary complex-valued function defined in D and l‘ is any selector of arcs, then there exists a selector of arcs [‘0 such that for each e19 e C, 1‘0(e19) is a finite or countable subset of F(ele) and nr(f,e19) = "1‘ (f,ele). (Gresser, 2, Theorem, 7, p.11) 0 Proof: Let y be any are of I“. Then BY = W - CY(f,ele) is open in W. 30 U BY = W - "P(f’ e16) and by the Lindelof covering prOperty there Yq‘ i is a countable subcovering with elements BYn of W - "F(t: e e) . Conse- °° 19 ie _ °° ie _ II 19 quently U B = W - n (f, e ) and H (f,e ) - n C (f,e )- r(f,e ). lhe_or_e_m _2_6_: If f(z) is an arbitrary complex-valued function defined in ’ i D and l‘ is any selector of arcs, then "P(f,ele) E Bflr(f,e 6) for all 19 except for at most a countable number of points e in C. (Gresser,2, Theorem 4, p. 6) 1129.9 For any positive integer j, let TJ. be a finite collection of compact neighborhoods on W which cover W and such that using the usual metric for W, we have diameter (G) < 1/3 for G any SUbset 0f TJ'. Choosing a finite number of G's for each j, we let Tj = 3 ij for each - ' i A i J and define P to be {e19 e C: "F(f’e 9) i B "F(f’e 9)}. Let _ 1 . 19 ¢}. P .-{e96P.annflr(f,e )7‘ [1,] 3 EaCh G . is contained in W - B "P(tflaiG) for each POSitive integer 3' n J If e16 e P, then there exists a point w 6 "F(f,e ) SUCh that 47 w a’ B “F(f,ele) which is closed in W. Thus there exist :1 and j such thatwe G . (\Bn (f,e19)= ¢. Therefore e16 e P , and P= U P nj 1‘ nj , a n,j naj. Now we wish to show that each Pn j is at most countable. In 2 order to show this, we fix j and n, and let e16 e Pn j' i If e16 is not 3 an isolated point of PH j’ then there exists a sequence { {k ’ . ie . d> in ij that converges to e . Since each {k e Pn,k’ ij finr(f,§’k)7‘ for each positive integer k. 80 for each #1 >0 Gr1 j n ufi(f,e16,u) ‘7‘ (P. i of points , Let {pk} be a decreasing sequence of positive real numbers which con- verges to zero. Then 16 = 00 7': 19 Gn,jnBflr(f,e ) kQI(an”"r(f’e ,uk))#¢ 9 , i because W is compact. This contradicts the assumption that e 6 6 PH j' 9 So each e19 e P . is an isolated point and the set P is at most :1 3 countable. Theorem 27: Let f be defined in D and 1‘ be a selector of arcs. If G M— ' i is any open subset of W such that for some e16 e C, GflB nr(f.e 9) 7‘ d” . 5-9 then there exists a sequence {(ji of points in C which converges to e such that CO HP(f,(,) 54¢ for each j. (Gresser, 2, Lemma 5, p.8) J This theorem follows easily from the definition of “I‘( ,e . £99393 1: Let f be continuous in D and I‘ be an admissible selector of . i arcs, For each point 819 e C, let 3 be an arc in I‘(e 6). Then BHP(f,eie) _C_ CB(f,eie) except for at most a set of first category. (Gresser, 2, Lemma 6, p.8) 48 Proof: Suppose the lemma is false. Then the set P denotes the set of ie t . 19 ie . points e e C for which B III.(f,e ) Q CB(f,e ) is of second category in C. Let 6 be an arbitrary positive number and S(ele,e) denote the set of all points in W whose spherical distance from CB(f,eie) does not exceed 6. Since CB(f,ele) is closed in W, it follows that for each point e16 e P there exists an G(ele) > 0 such that BIIP(f,ele) - S(elée) #Q Let i‘ji be a decreasing sequence of positive numbers which converges to zero and Pj {el6 6 P: Bl'lr(f,eie) - S(ele,ej) 7‘ 4’}. Since P = U P3 and is of second category, there exists a J such that PJ is of second cate- gory. We choose a finite collection {CPUUGm} of open sets each of diameter < eJ/4. For n g m let PJ(#) = {e16 6 PJ : Gfl(BHP- S(ele, e )) 7e 4’}. Since PJ is a union of the PJ(#)'s and PJ is of second category, there exists an M such that PJ(M) is of second category. For two subsets A and B of W, let a be any point in A and b be any point in B. Then the spherical distance X(A,B) between the sets A and B is defined to be the infimum of the spherical distances between points a and b. From the definition of PJ(#) it follows that X(G , Cfi)'>"3eJ/4 for any point e19 in PJ(M). According to Theorem 27 every point e19 in PJ(M) is a limit point of the set Q = {e19 e C : GMflHr(f,eie) 7‘ ¢}. We will now show that x_3eJ/4 for e19 e PJ(M). From the continuity of f’ K(GM:§) Z eJ/Z where S denotes the relative closure in D of 8. Let G be an open set such that f(S)C_ZG and x(GM,G) ZeJ/4. By the COhtihUity 0t f, the set U = f-1(G) is open in D and contains S. Since P iS an 49 admissible selector of arcs, there exists a subarc A_C_C such that each point of A is almost P*-accessible through S for I‘* the completion of I‘. So for every point eie e A, there exists an are a e I‘*(eie) such that aEU. Thus by the definition of U, C(f,e)EC for e19 e A. Since a. is a terminal subarc of 3, an arc in F(eie), the two arc-cluster sets are the same. Therefore, Hr(f,ele)§C—; for e19 e A and X(GM,HI.) 7/ /4, 6J a contradiction to the definition of Q. m g: Let f be a continuous function in D and I‘ be an admissible selector of arcs. Then III,(f,ele) = BIIl..(f,ele) for nearly every point e16 e C. (Gresser, 2, Theorem 8, p.11) 21319;; According to Theorem 25 for each e16 e C there is a finite or 19 is countable subset of F(e ), say aj(e ) such that . 0° . i H (f,ele) = fl C(f,ele,a.(e e)). F j=1 J If the set {aj(eie)} is finite, we repeat one of the arcs infinitely ' i often. For each j, let Pj denote the set {e19 e C : Bll g Caj(f,e 9)}. BY Lemma 7 each of the sets Pj is of first category in C. Therefore 1 i the set P = :J0 Pj is of first category and BIII.(f,e 9) E III.(f,e 6) for i j e9 E C - P. The proof is completed by using Theorem 26. i .. 19 31mg: If f is a continuous function in D, then “(f,e 6) -Bfl(f,e ) and II (f,eie) = BI'I (f,eie),where x denotes the collection of all chords 7‘ x i O at e 9, except for at most a set of first category. (Gresser, 2, Theorem 9 , p . ll) 50 Proof: In order to apply Theorem 28 we must show that A(ele) the col- lection of all arcs at e16 is an admissible selector of arcs. Let E be a second category subset of C and R(ele) e A(ele) for each e19 e E. For any positive integer j, we define Ej {e19 e E : A intersects the 00 circle lzl = 1 - l/j} . Then E = £1 E. and so there exists an N such 3'13 that EN is of second category on C. Therefore, there is an open subarc AEC such that EN is dense in A. Let S = U h(ele) where the union is taken over ei6 6 EN. Let G be an open set such that SEGED. We let e190 be an arbitrary point in A and Do be an Open disk centered at eie" having radius r S l - l/N. Let {in} be a sequence Of distinct points in ENDo which converges to e190. Then for each n, let kn be the component Of h(an) Do which forms a terminal subarc of K(fn). We will show that there is a component Go of GODO such that AngGo for infinitely many n. Let )tnk be a subsequence of kn's which converges to a limit set L. If LflDOflD 7‘ 4’, let 2 e LflDOflD. Then 2 e SflDo so that z is contained in some component C0 of GflDO. Since 2 e L(WGO and G0 is Open, it follows from the definition of limits that there exists an M such that Goflxnk7i ¢ for all k >M. Thus since Auk is a connected subset Of GflDO, Auk; GO for all k > M. So suppose LflDofl D = 4’. Let e19 e LflENflDO and a be the component Of K(eie)nDo which forms a terminal subarc of K(eie) . By the definition Of convergence, there exists an M such that aflknk 5‘ ¢ for all k > M. Since a is a connected SUhset of GOD , ais contained in a component C0 of GflDo. Furthermore, o Ink is a connected subset of GflDO. SO AnkgGo for k > M. Therefore, - t we have established that there eXists a component GO of'GflDo such tha AnSGO for infinitely many “- 51 For each positive integer k let D denote the open disk centered k at e190 having radius (1 - l/N) /k. Now we will construct a sequence {Gki of Open connected subsets such that GQG 2G2... and each G 9D . l k k If e16 e Dk’ let ak denote the component of aflDk which forms a terminal subarc of a. If e16 e Dk’ we let 01k = d>. Let itmi be a sequence of j~9o . distinct points in EN which converges to e Since there exists a component GO such that. inc; Go for infinitely many n, we can select an infinite subset T1 of {)4me and a component G1 of GflD1 such that ‘1’ 7‘ (11961 for each a 6 T1. Inductively we can define sequences {Gk} and {T } for each positive integer k such that Gk is a component of k GflDk, {K(fm)}2T12T ..., ¢¥ak§G for each are T . We fix k and 2 k k T . Since (19 '7‘ “kl-lgak, and atk-‘-1§G1(_|_1 and oszEGk ' d gGflDk. Since Gk+l is connecte let T , it follows H19- k that Std-106k 7‘ ¢. But Gk‘l'lanDk'l‘l and Gk is a component of GflDk which intersects Gld'l’ Gk-l'lEGk' Finally using the Gk's which are arcwise connected, it is possible to construct an are at eie" which lies in G. Consequently A(eie), the collection of all arcs at e19 is an admissible selector of arcs. Now we will prove that the theorem is true for X , the collection Of all chords at e16. Let {Aj} be a countable collection of closed Stolz angles at eie° = 1 such that each chord at e190 is contained in at least one of the Aj's. For each positive integer j, let Aj(eie) be the closed Stolz angle at e19 6- C Obtained by rotating Aj about the i . origin, Then for each j and each e16 e C, let Xj(e 6) be the collection 1 Of all chords at 819 which are contained in Aj(e 9). By an argument 1 . Complete1y analogous to that for A(e 6), Xj is an admissible selector 0f arcs. Consequently by Theorem 28 for each j there exists a set Ej 1 _ 16 19 c- E.. 0f first category in C such that anj(f’e 9) " ij(f’e ) for e E J 52 Since BHx(f,ele)EBIIX.(f,ele) for each j and ei6 e C, we have J i BHX(f,e e)EII _(f,eie) for e16 e C — E,. The set E = 3 E. is of xj ' J . .j=l J first category in C. So BIIX(f,ele)EIIX.(f,ele) for e16 E C - E. - JO Finally by Theorem 26 Hx(f,ele)SEBIIX(f,ele) except for at most a w . countable number of points in C, since BHXE .nIHX. = fix for e16 e C - E. J= J THEOREMS FOR SPECIAL TYPES OF FUNCTIONS As one might expect, there are numerous theorems relating to the theory Of cluster sets which are only valid for special types of func— tions. In the remaining sections Of this chapter we will consider some of the more important results for various types of functions including those which are continuous, light interior and locally univalent. Continuous Functions The et Qi curvilinear convergence _£,a function f is defined to be the set {x e C: there exists an are 7 at x and a point p in some . lim _ metric space such that 2.9.x f(z) — p . z e 7 Theorem 39: If f is a continuous function from D into W, then the set ‘— Of curvilinear convergence of f is a F05 set. (McMillan, 1,Theorem 5, p.302) First we wish to define special subsets F(n,j,k) of D. For each m a positive integer n let {A(n,j)}j=1 be an enumeration of the open disks each having its center at b, a point of W whose stereographic projection has rational real and imaginary parts and such that the set 53 {z e D: (f(z),b) < 4-n} contains points arbitrarily close to C. For each pair of natural numbers n and j, let {D(n,j,k)} be an enumeration of the components of the nonempty open set f-1(A(j,n))n{l - l/n 1 and integers jl’ j2, k1, k2 with the follow- ing properties. If eie e F(n,j1,k1) nF(n,j2,k2), then either A(n,j1)n A(n,j2) = 4) or there exist jo,k' and k" with k' 7‘ k” such that F(n,jpu E(n.j,>cA w, then w e CBl' Let A be an arc in D joining a point on 6 to a point on A so that the domain A bounded by 6, A and a subarc of C is a Jordan domain containing 1 in its closure. Let h be the restriction of f to A. Then C(h,l)= = CBl(f’1)° Since the preceding paragraph can be extended to Jordan domains, by conformal mappings there exists in A an analytic arc B end— ing at 1 such that CB(h,l) = C(h,l). Thus CB(h’1) = CB1(f,l). The are y can be constructed in a similar manner. If on and (12 are asymptotic paths of an arbitrary function f : D —> W l for the values a and a respectively, then d(a1,a2) denotes the infi- l 2 mum of rational numbers 5 such that some disk A, whose diameter is 6 and whose center has a stereographic projection with rational real and imaginary parts, has the properties (i){a1,a2}<:A and (ii) GI and a2 are eventually in the same component of f-1(A)n{l - é <|z|Aj+1 for J 2 1 and .QlAj -{a}. ponent U, of f-1(A_)r]{1-o, < '2' < l} and a positive integer nj such J J J that if n 3 n., then an is eventually in each Uj' Since Uj 2 Uj+1 for J > 1, there exists a boundary path a that is eventually in each Uj' / Since filA' = {a}, on is an asymptotic path of f for the value a and j= J p([an],[ oo- 57 M-Topology for Continuous Functions Suppose f is a continuous complex-valued function defined in D. Then we let T(p) denote the set of all Jordan arcs contained in DlJ{p} and having one endpoint at p, and let Gf(p) = {Ct(f,p) :t e T(p)}. In order to define the metric M, we choose two nonempty closed sets A and B in W and set M(A,B) = max(sug p and such that for each point z on any rectilinear segment [sj’sj+l] the condition d(f(Z),Kn) > / l/Zn is satisfied. Some subset of the union of segments Sj’sj+1] con- stitutes an element of T(p). Since Cs(f,P)n Ct (f:P)=:CS(f’p)n Kn: ¢’ n p is an ambiguous point of f. Corollary: Let f be a continuous function in D and E be the set of points p for which Gf(p) is not compact in the M-topology. Then E is a countable set. (Belna and Lappan, 1, Corollary 1, p.212) This corollary follows immediately from Theorem 33 and the Bagemihl Ambiguous Point Theorem (Theorem 4). Theorem 23: Suppose f is a continuous function in D and p 6 C- If {tn} is a directed sequence of arcs in T(p) such that Ctn(f,p) = KH and if K is a Continuum such that M(Kn’K) —) 0 but K é Gf(p), then there exists a directed sequence of arcs {8k }in T(p) and 6 > 0 such that for each is between tnk integer k > 0 there exists an integer nk > 0 such that sk a d ]] Iemma ] p.88) n t and d C f p),K) > e. (Lappan, a ’ nk'l'l ( Sk( , 59 2592:: We will prove this theorem by assuming that it is false and then showing that we obtain a contradiction. If this theorem is false, then for each positive integer k there exists an integer Nk such that for each 6 > 0 which is sufficiently small, n > Nk implies that all of the sets tnr1{z E D: 'z- pl < 5} lie in the same component of {z e D: d(f(z),K) < l/k, 'z-—p| < 5}. Therefore, for each n > N and each 6 > 0 k there exists a Jordan arc qn leading from a point of tn to a point of tn+l such that qnc:{z e D: 'z- p' < 5 and d(f(z),K) < l/k}. So we may choose a subsequence {tnk} of {tn} such that nk > Nk for each positive integer R. Then for each k there exists a Jordan arc pk leading from a point on t to a oint on t such that p c:{z E D: Iz- p < l/k and ”k p “H1 k | d(f(z),K) < l/k}, and the portion té of tnk between the terminal point 0f pk_1 and the starting point of pk satisfies the relationship M(f(t£,K) < l/k. Without loss of generality we may assume that pk meets tn and t in exactly one point each. Then letting t be the n k k+1 I Jordan arc obtained by splicing together all of the arcs tk and pk, we have Ct(f,P) = K contradicting the hypothesis K é Gf(p). IEEQEEE ii: Suppose f is a continuous function in D and p is a point in C such that Gf(p) is not compact in the M-topology. Then there exist directed sequences {tn} and {Sn} of arcs in T(p), e > 0 and a continuum K suCh that if Kn = Ctn(f,p) and Ln = Csn(f,p), then for each n > 0 M(Kn’K) < l/n, d(Ln’K) > 6 and the arc sn is between tn and tn+l' (Lappan, 11, Lemma 2, p.89) ' d‘tions flfiwf: Let{t%l}be a sequence of arcs in T(P) satisfying the con 1 re not Ctn(f:P) = Kn and M(Kn’K) < l/n Where K é Gf(p). If the arcs a 60 mutually disjoint, they can be shortened individually so that an infi- nite subset of the shortened arcs are mutually disjoint. If this was not true, there would exist an arc t e T(p) where t is contained in the union of the tn's and Ct(f,p) = K which contradicts the assumption on K. Now we can choose a directed subsequence of the tn's. In addition we can select an appropriate continuum K since Gf(p) is not compact. So the conclusion of this theorem follows from Theorem 34. Light Interior Functions A function f from D into W is called a light interior function if f is a continuous open map which does not take any continuum into a Single point. It has been shown that f has a factorization f = goh where h is a homeomorphism of the unit disk onto itself or onto the finite complex plane and g is a nonconstant meromorphic function. Let A(f) denote the set of all eie for which there exists an as- ymptotic path of f in D which includes e19 in its end and let Ap(f) denote the set of all ei6 for which there exists an asymptotic path of f in D which ends at the point eie. For any homeomorphism h of D onto D, we define B(h) to be the set of all e16 for which there exists an as- ymptotic path of h in D with end E and ei6 is contained in the interior of E. 1322£§m_29: Suppose f is a light interior function with factorization f = g°h - If A(g) is dense on C, then A(f)LJB(h) is dense on 0.. Fur- thermore, if Ap(g) and Ap(h) are dense on C, then Ap(f)LJB(h) is dense on C. (3. Mathews, 1, Theorem, p.79) 61 Proof: We will prove this theorem by assuming that it is false and show that we have a contradiction. Let the arc (w1,11,2)CC - A(f) be arbi- trary and [91,92]C(\y1,\y2) with 0 < 92 - 91<2Tr. Let I‘1 and 1‘2 be Jordan i6 16 arcs inD ending ate 1 and e 2 respectively with 1‘1“ 1‘2 = {0}. Then h maps the domain A bounded by 1‘1 UI‘2 and the arc [91,62] onto a domain A1 in D. Then there exist a point ela e Cr1(h,9 )0 Cr2(h,92) and sequences . . . id . {Zn} and {zn} in 1‘1 and 1‘2 respectively With h(zn) —-> e and h(zn) —> .a . e1 . Let A be a Jordan are at eLe which passes consecutively through the points h(zl), h(zi), h(zz), h(zlfl),.... According to Collingwood and Cartwright (Lemma 1, p.93), either [61,62]CCA(h-1,CL) or [62,61+2«n] C CA(h-l,a). Therefore, either (61,92)CB(h) or (62,91+2TT)CB(h) and Case (ii) [ZS-10 C]3Cr1(h,91) uCr2(h,92), with a proper inclusion. Then E = [510 CJ-[Cr1(h,91) Ucr2(h,92)] is a nonempty open subarc of C. Let eia be in bothEand A(g). Then e1C1 is in the end of an asymptotic path A of g. But C(h-1,a)C[91,62] so that h-1(A) is an asymptotic path of f whose end intersects [91,62]. Therefore, [91,62] 0 A(f) 7‘ ¢, a contradiction. Consequently both cases lead to contradictions. Since ($1,412) was arbitrary A(f) UB(h) is dense on C. The second part of the theorem is proved s imi 1 ar ly . 62 Locally Univalent Functions Any function f(z) meromorphic in D is called locally univalent if f(z) has at most simple poles and f'(z) # 0. The function has Koebe grgs if there exist curves JnCZD such that for some a < B < a + 2H and some constant c which is possibly x (i) Jn intersects the radii arg z =u and arg z = B for each n, (ii) IzI-—+ l for z e Jn as n-+ m, (iii) |f(z)- c| < e for z e Jn as n-—+ m. For any set C, the boundary of G is denoted by BC. Theorem 31: Let f(z) be a meromorphic locally univalent function with- out Koebe arcs. Then f(z) has three distinct asymptotic values on each arc of C. (McMillan and Pommerenke, Theorem, p.31) Erggf: Suppose that there exists an arc A of C on which there is at most one asymptotic value. So we may assume without loss of generality that f(z) has no finite asymptotic value on A. Let d(z) denote the radius of the largest disk around f(z) having no branch points on the Riemann image surface F. Since f is locally univalent there is a boun- dary point on the periphery of this disk. Seidel and Walsh (p.133) have shown that d(z) <;(1 - Izl2)'f'(z)' for 'zl < 1. There exists a se- quence {Zn} converging to some interior point g of A such that f'(z) is bounded. Consequently d(zn)-+ 0. Assume f(zn)-—+ c where c is pos— sibly m. Let Pn be the pre-image of the segment on F from f(zn) to the nearest boundary point bn' Thus f(z)-—+ bn for z e Pn as '2' —+~1. Since there are no Koebe arcs, Pn ends at a point, say Cn' lf(z)- f(znfl < d(zn)-+ 0, f(zn)-—+ c and zn-+ g for z 6 P11 as n-—+ w. 63 Then Cn—y g because there are no Koebe arcs on which f(z) --> c. Therefore, f(z) has the finite asymptotic value bn at gm 6 A. Now suppose there are no asymptotic values on the arc A except 0 and 00. From the preceding paragraph it follows that 0 and 00 are asymp- totic values on a dense subset of A. Let a e A be a point at which there is the asymptotic value 0. Hence there is a path P ending at a such that f(z) —-> 0 as z—> a for z e P. Let GO») denote the component of [z: lf(z)| < X, A > 0} that contains the part of P near a. Then C flaGOQCA for small positive )\ because there are no Koebe arcs on which f(z) —-> 0. For such a value of A, G(X) does not contain any as- ymptotic path for values 7‘ 0, but it does contain the path P on which f(z) —> 0. Since the Riemann image surface F does not contain branch points it follows that f(z) maps G(l) onto a copy of the universal covering surface of {O < le < A]. This construction can be performed infinitely often to obtain disjoint domains GkCD that are mapped by f(z) onto the universal covering surface of [0 < lwl é c{zn}(f,g) (iv) M*(f,A(§y\|11,\112)oo. From these sequences, we can ~ I - I = - I choose a sequence of pOints [zv} with Vl_1)mmzV g along which {f(zv)}-> oo in such a way that there is a corresponding sequence of points {Ev} on h(Cny) such that lim 0G ,z') = 0. By the application of véw V V condition (ii) we can choose a subsequence {znv] of [zv} so that f .., , , vl_1>mwo(znv,zv) oo V V V éoo V [f(zn )] is bounded. According to Gavrilov (l), there exists a p- v sequence {Ev} for the function f(z) lying in the non-Euclidean segments joining the corresponding pairs of points znv and 2“,. After some messy calculations involving bounds on lf'(z)l/(1+ 'f(z)'2) , condition (1V) is violated. So ; cannot be a Plessner point. 66 According to a theorem of Meier (Collingwood and Lohwater, Theorem 8.9, p.154), nearly all points of A are Meier points. For each Meier point, C(f,§) is a proper subset of W. So by Fatou's Theorem there exists a subarc Yo of Y around g with the property almost all of its points are Fatou points. Let FYo(f) denote the set of Fatou points on Yo. Then the set N0 = NFIFYO is a set of Fatou points whose linear measure is positive and its angular limit is a. From the Lusin-Privalov Theorem (Noshiro, l, p.60), f(z) E a CD aJ Theorem 3_9: Let g(z) = T‘- l - z where a is an integer > 4. _ l - a-J ' J' l 2flfl a LetAJ, 1' denote the disk with center at the zero Zj 1. = (l --T)e / ’ a for lI = 0,1,...,aj - l and radius l/(jZaJ). Then there exists a jo such that for all j 2 jo the interiors of the disks {Aj 1.] are disjoint, and a: g(z)—+ oo uniformly as z —>1 within D - (,Uj Li'AJ' 1.). (Krishnamoorthy, 1, J= o ’ Theorem 1 , p .94) Proof: Let z e D - (08 A, 1.) and 20 near G. Then there exists a k 0 a 1 0 such that l - a-k<|z I < l - a-k-l. We will decompose g(z) into four ‘ 0 products P1’ P2, P3 and P4 which we will specify below in order to obtain a lower bound of |g(ZO)|0 Let all k- 1 2 Th P 1(z)= Ugl -<———_—j) . en j—l l - a k-l aj k- 1 J 1<-1a,t(__1-_2_+1_1 [121(2)] Tl 1- >Tl 1-__a__'k -1 gea a > ‘1E>:UIJ l - a j j=l 1 " a j J=l \ 67 k a _2_k ) , which is holomorphic in the whole complex Let P2(z) = l -( l-a plane. Then ‘P2(z)| has its minimum in D - < 30 Aj 1.) on one of the j=j circles Ak,l' enclosing its zeros. Therefore, 0 k ak .I k ion . in k C ‘P2(zo)l= -1 (emu/,1, 28k ) -1 >e2“1’{1+e—2— ak L)+0(%k)?—§ ' k (a -l) k a - a k k+l k+l ( z )a '13 )l 20 a 3 Letting P (z) = l - _ _ , we have (z = —_—_ -l 3 l-ak l o l-a k l (k‘tl) m l aj The last factor P4(z) must then be fl [1 -( z_.) . Then k+2 l-a 3 co 2 aj 0° 1_ -k-l aj IP4(ZO)|=7T 1-( ‘3.) >H 1--a—:-— and k+2 l-a J k+2 l-a J -k—l aj j _ j—k-l (LL) < 4(1 - a-k-l)a < 4e a . Consequently I - a-J :18 u-l (I - 4e.a ) = C4 > 0. Therefore, 2 U 00 _ j-k-l IP4(ZO)| >W<1 - 4e a = k+2 C k-l lg(z )l > 2—L—2 (e1 - 2/a - I) which approaches a) as k +00. ° k (k‘l'l) Lappan (13) has recently used Theorem 39 to construct an example 0f two analytic functions f(z) and g(z) such that the spherical distance X(f(z),8(z)) +0 uniformly as |z| —> 1 and f(z)/é g(z). Let H(z) = fiH—m); 68 where s is a positive integer greater than 4. Using Theorem 39's nota- tion and conclusion we have that H(z) is an analytic function inD such 00 that H(z) +00 uniformily as lzl —> 1 in D - (_U. UA. I). We now want J=Jo J J, to construct an analytic function K(z) in D such that K(z) —-> oo uni- formly as |z| -+ l in ! and such that K(z) has no zeros in D. U A, jsl' 3’1 j l . ' '2 j S - For > 2 let D. = z :lzl 1 - 1 s 3 -2 s ) and A, = _U A, .. J / J [ < (/) /(J l J 1:0 1,1 By the Runge approximation theorem (Hille, l, p.303), there exists a sequence of polynomials [Pj(z)] such that for each integer j 3 2 le(z)I < (1/2)j for z 6 DJ. and lpzm + 123(2) +...+ Pj(z) -j|<(l/2)j . 00 for z E Aj. Setting L(z) = Z Pj(z), we have that L(z) is an analytic j=2 function in D and that for each j a 2, |L(z) — jl < l for z e Aj. Then K(z) = exp(L(z)) has the properties that K(z) -> oo uniformly as lzI—> l jLfi'AjJ' and that K(z) has no zeros in D. In addition |H(z)l2 + IK(Z)|2 -+ or) uniformly as lzl -> 1. Let f(z) = H(z)/K(z) and g(z) = in (H(z) ' 1)/l((Z). Then f(z) and g(z) are analytic functions in D, f(z) 7‘ g(z) and x(f(2),g(Z)) = 1/J[|H(z>|2+ lK|21 [1 + |g|21 . SO X(f(Z),g(z)) -> 0 uniformly as |z| -> 1. CHAPTER g NORMAL FUNCTIONS SUFFICIENT CONDITIONS FOR A FUNCTION TO BE NORMAL A family F* of functions f defined in a region Q is said to be normal if every sequence [fn} of functions in F* contains a subsequence {f } which either converges uniformly or tends uniformly toooon each compact n k subset of 9. A function f(z) is called normal in a simply connected region if the family {f(S(z))] is normal where S( 2) denotes an arbitrary conformal map of 9 onto itself. Noshiro (1, pp.87-88) cites the following conditions for a function to be normal. Theorem I: A non-constant function f(z), meromorphic in D, is normal if and only if a(f(z))|d(z)Lg Kdo( 2) holds at every point of D where d(f(z)) = lilfiéll_ d0(z) = _Jfl£l_ and K is a fixed positive cons- ’ 2 1+|fl2 1- lzl tant. (Lehto and Virtanen, l) gQEQlléEX: A non-constant function meromorphic in D is normal if and only if a(f(S(0)) is bounded for all conformal mappings S. (Lehto and Virtanen, l) 69 70 Theorem 2: Let f(z) be meromorphic in D, A(r,f) denote the spherical area of the Riemannian image of the disk |2| < r and L(r,f) denote the spherical length of the image of the circumference lzl = r. If A(r,f(S(z)).S KL(r,f(S(z)) for O < r < l where S(z) denotes an arbitrary conformal mapping of D onto itself and K is a fixed constant independent of S and r, then f(z) is normal in D. (Ahlfors, 1) Theorem 3: Let f(z) be meromorphic in D and A1, A2,..., Ah for q ;73 be mutually disjoint closed Jordan domains on the Riemann sphere. For j = 1,2,...,q, let pj denote the minimum of the numbers of sheets of islands of R above Aj where R is the covering surface generated by f(z). 1 T’>2’ q If there is no island of R above Aj, then #j ="*”- If .21(1- J= J then f(z) is normal in D. (Ahlfors, 1) QQEQAAEEXI A function f(z) meromorphic in D is normal if one of the following conditions is satisfied: (i) f(z) omits three values in D, (ii) the covering surface F has no univalent island above five mu- tually disjoint Jordan closed domains on W. (Ahlfors, 1) Other mathematicians have proved additional criteria for a function to be normal. These include the following results. IEEQEEE it A complex-valued function f(z) in D is normal if and only if I for each pair of sequences {zn} and [zé] in D such that 0(zn,zn) -> 0 the convergence of {f(z )} to a value a e W implies the convergence of n ( 7l [f(zé)} to a. (Bagemihl and Seidel, 2, Lemma 1, p.10) Since a normal function must be continuous, this theorem follows from a well-known result that a family of continuous functions in D is normal if and only if the functions are equicontinuous on each compact subset of D. (Hille, 1, Theorem 15.2.2, p.244) The sum of two analytic normal functions need not be normal as the next example will show. However, Theorem 6 will give a sufficient condition for the sum of two meromorphic functions to be normal. In order to construct two analytic normal functions whose sum is not normal (Lappan, 1, Theorem 3, p.190), we will first show that if f(z) is a normal holomorphic function in D, then for any two sequences . lim l = -f . [Zn] and {2;} in D such that 0(zn,z&) < M, n.—>«>f(zn) an 1 nli3;3f(q9 = m. If this conclusion is false, then without loss of generality we may assume that lim f(z') = 0. Let S (2) =(z + z;)/Ql + zgz). Since 11..)oo n n Sn(Z) is a linear transformation of D onto itself, the sequence of a . . = 0 functions [f(Sn(z))] forms a normal family. Since ilsm”f(§fi(0)) , the limit of the sequence {f(Sni(z))}, which we will denote by F(z), must be holomorphic in D. So there exists a positive constant L such that |F(Z)| < L in the disk o(0,z).S M. Then there exists a positive ' ' th 1nteger N such that f(Sni(z)) < L + l for all ni > N and all z in e ‘1 u = diSk 0(0,Z).$ M. However, setting Zn" = Sn (Zn)’ we have 0(0,Zn ) — " = f z . So 0(Sn(0),sn(znn))_ O(41,211) < M and f(Sn(zn )) ( n) 11 f " l to ”- i-qfim (Sni(zn1)) must be equa h 1 Now we will let f(z) denote a normal holomorphic function whic S i D such unbounded in D, and we will construct a Blaschke product Bf(z) n 72 that g(z) = f(z)Bf(z) is not normal. Let {zn"} be a sequence of points (X) ‘ 0° and Z (l -lz "l)<°°. We pick a subse- n=l n in D such that lim f(z ”) - Ii—rw n quence {Zn} of {zn"} such that for each j < n, G(zj’zn) > 3(n-j)M‘ where M' > M, the constant in the preceding paragraph. Then we choose I I = I = 0° J—zELI. zn—'z_ a sequence {Zn} such that G(zn’zn) M and Bf(z) “El Zn l-‘Enz . = = = I So Bf(zn) 0 and g(zn) 0. Since nlgm”f(zn) M, nlgmxf(zn) also equals 00. le(zr'l)l?_a> O by comparison with the Blaschke product in Example 4 of Bagemihl and Seidel (2, p. 11) . So lim g(zr'1)=oo and g(z) is not normal. n 00 Finally we define h(z) = 35(Bf(z) - 2)f(z) and G(z) = f(z) +h(z) . By direct verification h(z) is normal, and 6(2) =35f(z)Bf(z). A holomorphic function f(z) in D is uniformly normal if, for each M > 0 there exists a finite number K > 0 such that for each 20 e D, O(z,z°) < M implies that f(z) - f(zo) < K. If {Zn} and {2;} are two is close to sequences of points in D such that G(zn,zr'l) —+ 0, then {Zn} {2;}, or {Zn} and {2:1} are called close sequences. Suppose f(z) is meromorphic in D and there exist two close t" enma 1: sequences {Zn} and {21,1} such that f(zn) ->a and f(zr'1)—> B with all 8. Then for each complex number 6 with possibly two exceptions, there exists a sequence {2:} close to a subsequence of (Zn) such that f(zi) =6. (Lappan, 3, Theorem 4, p.44) M: Let Sn be a linear transformation of D onto itself mapping 0 into 2 and let Fn(z) = f(Sn(z)). Since St:1(zr;) -) 0, no subsequence of [Fn(z)} converges continuously at z = O and no subsequence of [Fn(z)} is a normal family in any neighborhood of z = 0. Suppose this lemma is false. Then there exists a neighborhood N of z = O and three complex 73 numbers a, b and c such that for each n in an increasing sequence of positive integers, Fn(z) omits a, b and c in N. However, by a theorem of Montel (Hille, 1, Theorem 15.2.8, p.248), this subsequence of func- tions is a normal family in N. Theorem 2: A uniformly normal function is normal. (Lappan, 3, Theorem 8, p.46) nggf: Let f be uniformly normal and {Zn} be a sequence of points in D such that [f(zn)} converges to a value a which may be infinite. Given M > 0 there exists K > 0 such that for each n 0(z,zn) < M implies that ,f(z) — f(zn)' < K. If a = “5 then f(zg) -)m» If a is finite, then {f(z$)] is bounded. So there exist three complex numbers 6i (i= 1,2,3) such that there is no subsequence {an-} of {2“} having the property . i that there exists a sequence {2:} close to [znk.} such that {2:} con- i verges to 61. So from the contrapositive of Lemma 1, f(zé) —+ a, and by Theorem 4, f(z) is normal. Theorem 9: If f(z) and g(z) are uniformly normal functions in D, then h(z) = f(z) + g(z) is uniformly normal. (Lappan, 3, Theorem 9, p.46) Proof: If M'> 0 is given, then there exist constants Kf and K.g such that for each 2 e D, O(z,zo) < M implies [f(z) — f(zo)I < Kf and .. = + ' /8(Z) g(zo)/ < Kg. Let K K? Kf Then for each 20 e D, U(z,zo)<:M implies [(f(z) + g(z)) - (f(zo) + g(zoD, < K. 74 Theorem 1: If u and v are harmonic functions such that f(z) =u(z) + iv(z) is uniformly normal, then u and v are both normal. (Lappan, 4, Theorem 6, p.158) Proof: Let [Sn] be a sequence of conformal mappings of D onto itself. Let 2n = Sn(0) and M > 0 be given. Then the family [Fn(z) = f(Sn(z) - f(zn)] is uniformly bounded in {z e D :0(z,0) s M/Z}. A subsequence [Fnk} can be chosen so that it converges uniformly on each compact sub- set of D. So {u(znk)] converges to a limit which may be either finite or infinite. Since Fn(0) = 0 for each positive integer n, F(z) = lim Fn (z) is a holomorphic function in D. If F(z) = U(z)+iV(z), k—>00 k then lim u S 2 = U z + lim u z ). If lim u z is finite, k....(nk(” <> k+w then u(Snk(z)) converges uniformly on each compact subset of D to U(z) + klimoouunk) while if kl_i)mwu(znk) =°°, u(Snk(z)) converges uni- formly to co on each compact subset of D. Therefore, u(z) is normal. Similarly v(z) is also normal. A special type of uniformly normal functions consists of the Bloch functions. A function f which is analytic in D is called a Bloch func~ tion if f(0) = 0 and it satisfies one of the following conditions: (i) sudef(z) < oo where df(z) denotes the radius of the largest z e single-valued disk with center f(z) on the Riemann surface f(D). (ii) sup (1 - IzI2)lf'(z)I <0... 2 e D + (iii) f(¢(z)) - fol/(0)) where w(z)=c1i+é . [H < 1. lCl < 1, form a finitely normal family where 00 is not allowed as a limit function. (1V) there exists a univalent analytic function g(z) in D such that f(z)=A logg'(z) for some constant A > 0. 75 Theorem 8: The above four conditions are equivalent. (Pommerenke, l, p.79) Proof: (i) ==>(ii): For any 21 e D we form 2 + z1 2 ‘k = — _ _ l r (z) [flll'flz ) f(zl)] [(1 21‘ )f (21)]. d (z) s. _,2_,_ > _ I (1 Izll >|f (21H (ii) =€>(i): From Schwarz's Lemma (Hille,lq Theorem15dul,p235) df(z)€ (l - lz’2)|f'(z)’ for '2' < 1 whenever f is analytic in D. ii .¢=> iii : T is o ows rom onte s eorem i e, , ( ) ( ') h' f 11 f M 1' Th (H'll 1 Theorem 15.3.1, p.251) and the fact that (1 -|z|2)lf'(z)| is invariant under w. (iv) =;>(ii): (l— Iz‘2)'f'(z)l=l(l -lz|2)'§;%§%l, which is bounded. (Hille, 1, Lemma 17.4.1, p.351) ,, , . _ 2 g”gz) . (11) ==> (1v). 'zpqp1(l Izg )’8'(Z)I < m, which implies g(z) is univalent by Nehari (1). If f(z) is analytic in D and f'(zo)/¥ 0 for 20 E D, then the maxi- that is mapped by f(z) one-to-one onto a mal domain containing Z0 Gross single-valued domain starlike with respect to f(zo) is called the domain G(zo) of f. Rays of G(zo) are defined to be the preimages star If R is a ray of G(zo) then of the rays of the starlike image domain. either R is a Jordan arc in D that goes from 20 to a point 21 e D where f'(§1) = 0 or R is a Jordan arc in D except for its endpoint e 9 e C, where f(z) has an asymptotic value. 76 Lemma _2_: If f(z) is analytic in D and without Koebe arcs, then for any 19 point e e C either f(z) has an asymptotic value at e19 or diameter G(z)—>0 as z—> e19. (Pommerenke, 1, Theorem 7, p.90) m: Suppose G(z)—#0. Then we have {zn}—->eje with dia G(zn) >/ro >0. So there exist rays Rn of G(zn) such that die Rn >/ro. We have two cases. (i) There exists a subsequence [nk] and some r, 0 < r < r0, such that min {'21: Iz— eiel < r, z E Rnk} -—>l as k-—>00. Then some subarcs Rfik converge to an open arc A0 of C that has eie as an endpoint. In addition f(Rfik) is either a line segment or a half line. We claim that f(zo) is analytic on A0. We may assume without loss of generality that the endpoints of the segments f(Rfik) converge respectively to the points WI and w" in W. Furthermore, we may assume that the directions of the segments f(Rnk) converge. So as nk->oo f(Rnk) converges to a recti- linear segment L joining w' and w" (which may be the same point). By a suitable linear transformation we can make L a real segment or a single point. Let g1 and g2 be distinct points on A0 and choose points 21; and II zn on Rfik such that zr'l—rgl and 23+QZ. Without loss of generality we may assume that the corresponding sequences of points f(zh) and f(zg) converge. Neither of these limits can be 00 since f(z) maps Rg‘k one-to-one onto f(Rnk) and f(z) has no sequence of Koebe arcs for en. Therefore, by replacing AD by its are between Q1 and {,2 we may assume that L is bounded. We will now show that f(z) is bounded in a neigh- borhood of each point of A0. Suppose on the contrary that there exists a point Q3 6 A0 and a sequence of points zj e D such that 23 —>§3 and f(zj) +00. Let Lj denote the half-line {Tf(zj) : T >1} and let A3. be the component of the preimage f—1(Lj) that contains zj. We choose zj's 77 such that f'(z),¥ 0 on Aj. Then Aj is a simple curve tending at one end to a point of C. For all sufficiently large j, Lj does not inter- sect Uf(R&k) because L is bounded. So f(z) has a sequence of Koebe arcs for m, a contradiction. Consequently f(z) is analytic on A0 and has an . i asymptotic value at e 9. (ii) We can find c1 6 D such that a subsequence of Rn comes arbi- If (i) does not hold, we can also find c e D trarily close to c 2 1. with 0 < lcz -eie| < lc1 -eie|/2 and such that another subsequence of Rn comes arbitrarily near to c2. After continuing this process of taking subsequences we finally take the diagonal sequence. So we have k e D with ck —+ e19 as k —+ «>such that for each fixed k, Rn comes arbitrarily close to ck as n —>m. Since f(Rn) is a line segment points c or a half-line, all the points w = f(ck) lie on the same line L. The k points ck are all distinct since f(z) is not constant. Since f(z) maps Rn one-to-one onto f(Rn), wk converges monotonically along L to a limit wo. Let Dk be a disk around ck such that diameter f(Dk) —+ 0 as k —> m. For each k, we choose nk k k k + + Dk+l' Then Al-+ [gl,b1] [a2,b2] ... may be assumed to be a Jordan 9 and a subarc Ak of Rnk from a E D to b e , i arc that lies in D except for its endp01nt e f(Ak) converges to [hk_1,wk]. Since wk —> W0 and diameter Dk —+ 0, f(z) has an asymptotic i value w at e 6_ 0 Theorem 2: Every Bloch function in D has finite or infinite angular limits on an uncountably dense subset of C. (Pommerenke, 1, Theorem 8, p.91) Proof: Since every Bloch function is normal, every asymptotic value 78 is an angular limit. Let A be an arc of C. Suppose there exists an interior point e16 of A where there is no asymptotic value. By Lemma 2 there exists a Gross star domain G(zo) such that Boundary G(zo) CiA. The number of rays of G(zo) has the power of the continuum and only countably many of them can end at points where the derivative is equal to 0. All others end on C. Therefore, it is sufficient to prove that any two distinct rays end at distinct points of C. Let R1 and R2 be different rays of G(zo) with endpoints Q1 and g2 in C. Since af(z) + b is also a Bloch function, we may assume that the half-lines or segments f(Rl) and f(RZ) lie on different sides of the line {Realw = Realf(a)]. f(z) So the normal function e tends to different limits along R and R2. 1 Consequently C1 f C2. Theorem lg: Suppose F(z) is the Blaschke-Quotient expressed in the form F(z) = B1(z)/B2(z) = n1 8 Wand Z (1 -lb ') < «n If the set of limit n=1 n n=1 n , points of the an's is disjoint from the set of limit points of the bn's then F(z) is normal. (Cima, 1, Lemma 1, p.769) Proof: lBl(Z)Bé(Z) ' Bi(Z)B2( z)|(l - I212) < lB' 2(z)J(1 - ‘2}2)+ ‘Bl(z)\(1 - \Z‘Z) ‘Bl(z)|2 + |32(z)l2 l31(z)|2 + \32(z)\2 ie lifl(r31(2)‘24"B2(z)|2) 3,1 as z-—+ e in D and |Bi(z)‘(l- lzl) for l = 1,2 is bounded for [z] < 1 according to Seidel and Walsh (1). So I _ v Ii_m_|131(z)132(z) B1(z)32(z)| 2 lBl(z)l2 + le(z)| _lzlz) < m as z--)-ele in D and 79 'Bl(z>Bé(z) ‘ Bi(Z)Bz(Z)'ldzi cldzl ll= d(F(2)) dz 'Bl(z)'2,_'32(z),2 < (1 - IZIZ) the condition in Theorem 1. CLUSTER-SET THEOREMS FOR NORMAL FUNCTIONS The following theorem of Lehto and Virtanen is one of the first important results in the theory of cluster sets of normal functions. Theorem 11; Let f(z) be a normal meromorphic function in D. If f(z) has an asymptotic value a at a point 20 on C along a Jordan curve lying in D, then f(z) possesses the angular limit a at 20. (Lehto and Virtanen, 1, Theorem 1, p.49; NQshiro, 1, Theorem 6, p.86) Bagemihl and Seidel have proved many other cluster-set properties of normal functions. These include conditions for f(z) to be identi- cally constant and conditions for f(z) to have a limit at a point. Theorem 1;: Let f(z)be anormalmeromorphic function in D Which omits the finite or infinite value c and let (Zn) be a sequence of points in D which converges to a point i e C. If there exists a positive number M such that for every n, 0(z ,z + ) < M and if lim f(z ) = c, then f(z) n n 1 n -+oo n has the angular limit c at g. (Bagemihl and Seidel, 2, Theorem 1, p.4) 2 + z lq-Ehz for n any positive BEERQ: The family of functions gn(z) = f integer is normal in D and lim gn(0) = c. So the functions [gn(z)} [1900 80 converge uniformly on every compact subset of D to c. Let S be the compact subset '2’ $ A where l > A > tanth Since G(zn,zn+1) < M, the um 1- l in its interior. lim f(z) = c when 2 is re- 2 —*§ non-Euclidean circle Ah with center 2n and radius equal to %log contains the pOint zn+1 stricted to the union of the interiors of the circles Ah' In particular this relation holds if 2 lies on the polygonal line formed by joining the points 2n and Zn+ by a Euclidean line for all n. So f(z) possesses 1 the angular limit c at g by Theorem 11. A boundary path is a simple continuous curve 2 = z(t) (0 S t < l) in D such that lz(t)| ->l as t -+1. The initial point of the boundary path A is the point 2(0) and the end E of A is the set of limit points of A on C. In order to decide when two boundary paths are ”close toge- Il * = . * = . ther , we let D1(A1,A2) tliml sup 0(z(t),A2),D (A1,A2) tliml sup z(t)eA1 z(t)eA2 * = * * 0(z(t),A1) and D (A1,A2) sup{D1(A1,A2),D (A1,A2)]. If P is a prime end of D, {gm} is a chain belonging to P and dn is a I Q o _ = —' the subdomain of D defined by qn and containing qn+l’ then rldn Fldn for {qé} any equivalent chain. The set I(P) = erg, which is invariant in the equivalence class of chains constituting P, is called the impression of the prime end. Two distinct prime ends of D can have the same impression. For example, if the domain D is obtained by deletion Of an end-cut y from the unit disk, then each interior point of y cons- titutes the impression of exactly two prime ends. Theorem 13: Let f(z) be a non—constant meromorphic function in D that \— tends to C along a boundary path A whose end E contains more than one 81 point. Then given E>>O there exist boundary pathsA1 and.A2 whose ends are contained in E such that A”.A1 andA2 are mutually exclusive; D*0\1,A2)<:e; and f(z) —*c along.A1, but not along A2. (Bagemihl and Seidel, 1, Theorem 1, p.264) 2322:: Let G = D -.A. The initial point of A.is the impression of one prime end of G whereas every other point of A.is the impression of two prime ends of G. If E is the impression of a prime end of P and E = C, then E is the impression of only P, but if E.f C, then E is the impres- sion of P and of another prime end P'in G. If E = C, we map G onto D in a one-to-one conformal manner so that the initial point of A and the prime end P correspond respectively to the points -1 and 1. Let F(z) denote the image of f(z) under the con- formal mapping. Since f(z) ; c, there exists a sequence of points in G tending to C on which f(z) —>b # c. So there is a sequence of points in D tending to the point 1 on which F(z) —>b and there exists a segment S in D bounded by a suitable arc and chord of C both having an endpoint at 1 that contains infinitely many points of this sequence. F(z) —>c as z—>l along C but not as z —>l on S. Consequently from an argument of Lehto and Virtanen (1, pp.49-52), given 6 > 0 there exist two disjoint boundary paths Ai and A5 in D whose ends are the point 1 such that D*(A1,A2) < e and F(z) -+c along Ai but not along Aé. So under the original mapping of G onto D there exist boundary pathsA1 and A2 that lie in G and satisfy the conditions of the theorem. If E f C, then we map G onto D one-to-one conformally so that the initial point of A and the prime ends P1 and P2 correspond respectively to the points -1, -i and i. Let F(z) again denote the image of f(z) 82 under this conformal mapping. Let A, A1 and A2 denote the open subarcs of C which, when described once in the positive direction, have the respective initial and terminal points -i and i, -l and -i, i and -1. Under this conformal mapping A corresponds to the arc C-E while Al and A2 each corresponds to A minus its initial point. Therefore, as the point i or -i is approached along A, the inverse of the mapping function approaches an end point of E. Then this limit is approached as z-+i or -i on the set {lzl s 1, Real( 2) 1;,A}. Since f(z) é c,according to PriValow (l, p.207) there exists a sequence of points in G tending to an interior point of E on which f(z)—+ b # c. So there exists a se- quence of points {Zn} tending to i or -i satisfying the conditions: Real(zn)-<0 for n any positive integer and F(zn)-+-b as n-+om In add— ition F(z) —>c as z —>-i along A1. The rest of the proof is the same as above. Theorem lg: Suppose thatf(z)is.anormal meromorphic function in D and that Al and A2 are boundary paths for which D*(A1,A2) is finite. If f(z)—+.c along Al, then f(z) ~>c along A2. (Bagemihl and Seidel, 1, Theorem 3, p.266) Proof: Assume c is finite. If C=cw, then we will look at the normal meromorphic function l/f(z). If this theorem is false, then there exists a number c: possibly w,different from c and a sequence of points {2'} on A such that 11m Iz‘l = l and lim f(z') = c'. Since D*(A ,A ) n. 2 n+°° [1 11+” n 1. 2 is finite, there exists a positive number M and a sequence of points [Z } on A such that lim Iz 1 = l and 0(z ,z') < M for n any positive n l ,,_+wn n n n integer. The family of functions [f(Sn(z))} where Sn(z)= (z+zn)/(1+Enz) 83 Since c is finite, is normal in D. As n—rao, f(Sn(0)) = f(zn) —>c. there exists a subsequence [f(Snk(z))} which, as k-94n, converges uni- formly to a meromorphic function F(z) on the closed disk A in D whose center is the origin and whose non-Euclidean radius is M. For all suf- ficiently large values of k, 8;:(A1) intersects every circle O(0,z) = L with L < M. Since f(z) -+c.along A1, F(z) 5 c. However, G(znk,zfik) < M so that 8;:(zék) e A and since f(zfik) —>c' f c as k.—>ag F(z) é c, a contradiction. If a normal meromorphic function f(z) in D tends to a limit Corollagy: along a boundary path whose end contains more than one point, then f(z) is identically constant. (Bagemihl and Seidel, 1, Corollary 1, p.266) In the section on Locally Univalent Functions in Chapter I we de- fined Koebe arcs of f(z). A Koebe seguence pf arcs relative 59 pp open app A of C is a sequence of Jordan arcs [Jul in D such that (1) for some sequence {en} satisfying the conditions 0 < En < 1 for n any positive integer and En —>0 as n+oo, Jn lies in the en-neighborhood of A, (11} every open sector A of D subtending an arc of C that lies strictly interior to A has the property that, for all but at most a finite number of n's, the arc Jn contains at least one Jordan subarc lying wholly in A except for its two end points which lie on distinct sides of A. Let f(z) be a normal meromorphic function in D. If f(z)—>c Theorem 12: along a Koebe sequence of arcs {Jn}, then it is identically equal to c. 84 (Bagemihl and Seidel, 3, Theorem 1, p.10) If c is a finite non-zero complex number, then we Proof: Assume c = 0. replace f(z) by f(z)- c;or, if c = «5 we replace f(z) by l/f(z). Let {Jn} be the given sequence relative to an arc A. We define an arc B = {z :lz' = l, q1 < argzz < qz} to be strictly interior to A. We denote by A the open sector of D with vertex at the origin and vertex angle 6 subtending the arc B. There is no loss in generality in assum- ing that for every n the arc Jn contains a Jordan subarc {h lying (1) and Piz) which lie on the wholly in A except for its endpoints Pn max |21 for n any sides . = min d R s1 and s2 of A We set rn ze Pn‘z' an n ze rn positive integer. Then lint rn = lim Rn = 1. Now we define a Jordan n+oo 11+!” 1 and 32 respectively at curve Kn for each n. Let ‘2' = Rn intersect s (1) (2). If Bn is the open arc of lzl = Rn which lies the points Qn and Qn in A and B: is the complementary arc, then we define Kn to be the union of P(1)Q(1) 8*, P(2)Q(2) and P . An argument involving harmonic mea- n n n n n n sure shows that if D is mapped conformally onto the interior of Kn by Z = ¢n(w) where ¢n(0) = 0, then for n sufficiently large the arc [h is the image of an are S on C having a length at least F times the harmonic measure w(O,Bn,{z :lz’< RHJ). Since f(z) ->0 along the Koebe sequence {J 1, lim f(¢ (w)) = 0 n n—)oo n uniformly on S. From Lehto and Virtanen (l, p.64),{f(¢n(w))] tends to zero uniformly on every compact subset of D. Suppose there exists a point 20 e D for which f(zo) is not zero. is in the inter- BY the definition of a Koebe sequence relative to A, z = ¢:(z) denote the in- ior 0f each Kn for n sufficiently large. Let w Verse of z = ¢n(w). Then f(¢n(¢:(zo))) = f(zo) for n sufficiently 85 large. Since {f(¢n(w))} +0 uniformly on every compact subset of D but f(z ) 7‘ 0, lim '¢*(z )l = 1. However, if p is fixed so that lz |l5 'zol/p < 1 for n suffi- ciently large, a contradiction. Theorem lg: Let f(z) be a normal function in D that omits the value w If which is either finite or infinite and let A be an open subarc of C. the set of Fatou points of f(z) on A is of measure zero, then A contains a Fatou point of f(z) at which the corresponding angular limit of f(z) is w. (Bagemihl, 1, Theorem 1, p.3) If f(z) were bounded in some neigh- Proof: Assume w is 00. Let g e A. borhood of Q, then by a simple extension of Fatou's Theorem, the set of Fatou points of f(z) on A would be of positive measure, which is con— trary to the hypothesis. So f(z) is unbounded in every neighborhood of g. Hence there exists a number 5 > 0 such that the region H = D n {z ’2 ' Q! < 5} satisfies the conditions that H0 C C A and f(z) is un- Consequently there exists a sequence of points {Zn} in D (M <...< bounded in H. l 2 such that zn -+§ and Mn = 'f(zn) 1 ->oo as n+oo where l < M For 11 any positive integer, let Vn be the open set of points M n <. . . . Let Rn denote the component of Vn that in D for which 1f(z)l > Mn ~ 1. contains zn [f(z)! = Mn - 1 at all boundary points of Rn that lie in D By the maximum principle, Emma is non-empty. Suppose the diameter __ miE Let rn - 26erlzl. Since f(z) of Rn does not tend to zero as n->oo. l and there exists a Koebe se- omits oo in D b a um tion lim y 38 P ’ n+oo Quence of arcs along which f(z) +00, a contradiction of Theorem 15. rn Thus there exists a natural number N such that RNCH. 86 We want to show that f(z) is unbounded in G1 = RN' Let G? be the smallest simply connected region containing G1 and z ? ¢(w) be a func- tion that maps D conformally onto G*. The set B* = G%FWC is non-empty. 1 We denote by Bf the set of all points of B* that are accessible from CY. 111 _ lim ill . . . . let ¢*(e )‘- r_+_1 ¢(re ) for every u for which the limit ex1sts. By Fatou's Theorem this limit exists at almost every point of C. The set E1 = (sin: '¢*(elu)' = l} is a Borel set and hence measurable. In addi- iu tion B? = {¢*(elu) :e GEE We want to show that the function g(w) = l]' f(¢(w)) is unbounded in D. Assume not. Suppose m(E1) > 0. Let EO denote the Borel set of positive measure which is the subset of E1 consisting of all the points for which g(w) possesses a radial limit and B: be the image of EO under the mapping 2 = ¢(w). From an extension of Lowner's Theorem (Tsuji, l, p.322), B: is a measurable subset of Bf with m(B:) > 0. Let goeng. Then there exists a path in G? terminating at £0, and this path is the image under Z = ¢(w) of a path in D that terminates at a point e1u° L + 1 is non- empty. Let G2 denote a component of this set and f(z) is unbounded in G2 as before. A continuation of this process yields a sequence of . D ‘ .— nested subregions G1 G2 I) of H. Now we choose zleGl, 22 6 G2 {21}, z3eG3 - {21,22}, . . ., zneGn - (21,22, . ..,zn_1], . .. and jOin 21 to 22 by means of a Jordan arc J1 lying in G1. In addition we join z2 to 23 by a Jordan arc J2 lying in G2 and having no point except z2 in com- mon w1th J1,..., jOin zn to zn+1 by a Jordan arc Jn lying in Gn and having no point except Zn in common with J1 U J2 U... UJn-l’ . So 00 P = U1 Jn is a path in D with initial point 21. Its end lies on C n: because hm m1“ n-> ‘f(z)|= 0° and f(z) omits 00 in D. According to the co ZEJn Corollary following Theorem 14, the end of P is a single point :6 C. Since f(z) is normal in D, g is a Fatou point of f(z) with the corres- ponding value 00 by Theorem 11. In conclusion if w is finite, we then define F(z) = m . From the proof above, A contains a Fatou point of F(z) with the corresponding angular limit 00. So this is a Fatou point of f(z) with the angular limit w. A hypercycle is the locus of points whose non—Euclidean distance from a given non-Euclidean straight line is constant. m l_7: Let f(z) be a normal meromorphic function in D that omits the finite or infinite value w. If there exists a sequence {Zn} having at least the limit points a and B on C and a constant M such that G(zn,zn+1) < M for every n and nl_j-.;noof(zn) = c, then c 5‘ w and f(z) E c. 88 (Bagemihl, 1, Theorem 2, p.4) Ppppf: Assume c = w. Then by an argument in Theorem 12's proof, there exists an asymptotic path P in D whose end contains the arc a8 such that z¥3?1 f(z) = w. According to the Corollary following Theorem 14, this fmilies that f(z) E w, a contradiction. Assume f(z) i c. If the set of Fatou points is of measure zero, then by Theorem 16, since f(z) omits w, f(z) has a Fatou point on the are as at which the corresponding angular limit of f(z) is w. If the set of Fatou points is of positive measure, then by a theorem of Privalow (Noshiro, 1, Theorem 2, p.72), f(z) has a Fatou point g on the arc QB at which the corresponding angular limit of f(z) is d # c. Let Y be an angle such that 0 < Y < gfi and M < logtanQn-+ py) where M is the constant in the statement of this theorem. Let A be the subregion of D determined by the two hypercycles that form the angles y and -y at g with the diameter of C joining g and -§. In a neighborhood of g every point of A lies in a symmetric Stolz angle of opening ZY. So :hgégf(z)= d r c. Since nEEEnf(zn)= c, the points 2H for all suffi- ciently large n do not lie in A. Since every point of a6 is a limit point of {Zn}, for infinitely many n the points zn and zn+1 lie on op- posite sides of A. Every boundary point of A that lies in D is at the non-Euclidean distance plog tan(%fi'+ %Y) from the diameter of C joining ‘C and g. Therefore, for infinitely many n, G(zn,zn+1) Zzlogtan(%fi + hY) > M, a contradiction. 89 SUBHARMONIC NORMAL FUNCTIONS One special class of normal functions, the subharmonic normal functions, have the property that for any u for which IT1u(reie) I = 0(1), 11 has Fatou points almost everywhere on C. A continuous function u(x,y) not identically equal to zero is £12- harmonic if and only if it satisfies either of the following mean-value inequalities for each circular disk in D: u(x y ) < ‘1'- 211u(x + ecose y +Psin9) d9 0’ o \ 2T1 o o ’ o ’ u(x y ) < Lfr fznu(x + 9cose y + Psine)9 d9 d6- 0’ o \ 2 o o o ’ o Trr If u(x,y) has continuous second partial derivatives in D, then it is subharmonic if and only if 2 2 Au=a_121+.a_;> 6x 3y 0 at every point of D. Theorem Q: If u is normal and subharmonic in D and f:"1u(reie)ld6 = 0(1) for Os r < 1, then u has Fatou points with finite Fatou values almost everywhere on C. (Meek, 1, Theorem, 13-314) M: According to Littlewood (1, Lemma 3, p.390) 11 has the represen- tation u = v + u* where v has the property that if w9(z) is harmonic in ._ .. lim _. . z _ ._ 4,. l l < 9 < 1 and W9 u on [Z] 9 , then 9 1W‘.(z) v(z) and u is a . . . . . i9 non-p031tive subharmonic function in D With u*(re )+ 0 as r +1 for 90 almost all e 6 [0,2"). In addition by a theorem of Tsuji (1, Theorem 1V.16, p.147) v has Fatou points corresponding to finite Fatou values almost everywhere on C. Let E denote the set of points on C at which u* has radial limit as zero and v has a finite Fatou value. For any 8, 05 B < “/2, and e:L E, 1et H(eie,B) denote the open set in D bounded by the hypercycles from --e19 to e16 making angles 8 and -B respectively with the diameter . . a, . through e19 and -ele. We pick a sequence {Zn}n=l S H(ele,B) such that i zn-+>e 6 as n-auw. For each positive integer n, we denote the non-Euclidean straight line which passes through zn and is perpendicular to the radius 9e16, 0.3'0 < 1, by En' Because of the invariance of the metric 0 under one- to-one conformal mappings of D onto itself, it can be shown that each of the bounding hypercycles of H(e16,8) is at a hyperbolic distance 0(O,tanB/2) from the diameter between e19 and -ele. Therefore, for i . . each positive integer n, one 6, the point of intersection of En with 9e19, satisfies the relation 0(9neie,zn) S o(0,tanB/2). For each positive integer n,8n(w) = (W“+ pneie)/(l +’9ne-iew) is a one-to-one conformal mapping of D onto itself. Since u is normal, there exists a subsequence, also denoted by {U(Sn)};:1, which converges uniformly or diverges uniformly on the com- pact set K = {w:0(0,w) g O(O,tanB/2)}. Since u(sn(o)) .-_- u(Pneie) = v(eneie) + u*(9neie) —>v(6), the Fatou value at cm, the subsequence cannot diverge uniformly on K. 30 the subsequence converges uniformly on,K to a subharmonic function U. We have u(Sn(w))<$'v(Sn(w)) for w e K and any positive integer n. 00 Since e16 is a Fatou point of v, [v(Sn)}n=1 converges uniformly on K 91 < =1' = to v(e) and U(w) \~v(6) for w e K. But U(O) n-:F&F(Sn(0)) v(e) and by the Maximum Principle for subharmonic functions U(z) E v(e) in K. Furthermore, E has linear measure 2“. So u has Fatou points almost everywhere on C. Corollagy A: Any normal subharmonic function on D which is bounded above has Fatou points with finite Fatou values almost everywhere on C. (Meek, 1, Corollary 1, p.316) 3529;: If u is normal, subharmonic and bounded above in D, then v==eu is also normal, subharmonic and bounded in D. In addition every Fatou point of v is a Fatou point of u. So u has Fatou points almost every- where on C. By Arsove (1, Theorem B, p.260), a subharmonic function bounded above on D has finite radial limits almost everywhere on C. Consequently u has finite Fatou values almost everywhere on C. Corollagy B: If u is normal, subharmonic, bounded below and admits a harmonic majorant v on C, then u has Fatou points with finite Fatou values almost everywhere on C. (Meek, 1, Corollary 2, p.316) Proof: Without loss of generality we may assume that 0.< u(z) for zeIL Then 0 5 I:Wu(rele)d6\< fjflv(rele)d6 = 2mm) for o\< r < 1. So Theorem 18 applies. In order to generalize Theorem 18, the following questions must be answered: Must a normal subharmonic function in D have any Fatou points on C? If so, is this set dense on C? 92 BOUNDARY BEHAVIOR OF NORMAL FUNCTIONS If A is any Stolz angle at eie, we define HA(f,eie)=C]CT(f,eie) where 1' is any simple continuous curve in A and RA(f,eie)=E*int RA*(f,eie) where KA*(f,eie) is the range of f in any Stolz angle A? that strictly contains A. A function f has the p—segment property for any integer n 3.2 if there exist n chords F1,...,Ih terminating at eie such that Crff,ei%fl... (\CP(f,ele) = o. In this section it will be shown that for any normal n . . . . i6 . meromorphic function f in D the set of pOints e at which f possesses the n-segment property is of first category and measure 0 on C. Theorem 12: If f is normal and meromorphic in D, then for any e19 e C and any symmetric Stolz angle an of opening 2d at e19, CA (f,eie) - EA (f,eie) e IIA (f,eie). (I U. a (Rung, 1, Theorem 1, p.44) Proof: Let c be any arbitrary point in 9A (f,ele) - EA (f,ele). Then G. O. i6 - there ‘ ' A ' h ' = 11m = ex1sts a sequence {Zn} in (I Wit nlimwzn e and n.~>«fi(zn) c . i -' Let wn be the unique point on the diameter of C from e 9 to e 16 for which U(zn,wn) equals the non-Euclidean distance of zn to this diameter. For any 3 satisfying a < 3 < n/Z, let HB denote the region bounded by the two hypercycles symmetric in this diameter that form the angles 8 and ~B with it. If Bl is chosen so that a < Bl < B, then there exists a positive integer N such that n > N implies U(zn,wn) < s log cot(3zTT- #31) = M (l) 93 because -—— l n a ' < ... _ _ _ n11m U(zn,wn) \ 2 log cot(4 2) . + w We define gn(§) = f(Igngflzfi for any positive integer n. Since f n is normal, there exists a subsequence {gnk] which converges uniformly in any compact subset of D to g. We also define another sequence {Ck} by gk + nk e2M - l = <-——= “k 1 _hka . Because of (l) ICkl eZM 1 K, and so for any accumulation point Co of {§k], |§o|.S K- the equations z Let {gp} be a sub- sequence of {§k] tending to go Then the continuous convergence of{gnk} im lies that lim ( ) = lim f(Zn ) = g( ) = c. P p —>oognp Cp p-koo P go We want to show g(g) E c. Suppose not. Let D* be any fixed non— Euclidean open disk with center go which is contained in the open disk 1 e2 * - l 1 1 B _—__ i: = — —_— 1C1< 2 e2M* 1 where M 2 log cot(4 2). Let 3 be any point in D*. By Hurwitz's Theorem (Caratheodory, 2, p.195) there exists an integer K0 and a sequence of points {3k} in D* that tend to 3 such that gnk(3k) = g(z) for all k.;’Ko' For the points xk = +W _ ' = . i6 (3k nk)/(l+3kwnk) for which kgKo, f(xk) g(z) So g(z) elifi(f,e ). Since g( g ) i c, cesEA (f,ele), a contradiction. Consequently g(g) a g (I. which implies that f tends uniformly to c on the sequence of non- and radii M. Each disk intersects both Euclidean disks with centers Wnk 6). boundary segments of Ah; so CEIL§(f,el a In Chapter I we defined K(f) to be the set of points §E'C for which QA1(f,§) = QA2(f2C) for any pair of Stolz angles at g and the outer an- gular cluster set CA(f,§) to be the union of all cluster sets q3(f,§) for A any Stolz angle at €- 94 Theorem 20: If f is normal and meromorphic in C and geK(f), then, for any Stolz angle A and any chord (11((1) terminating at g and making the angle a for "Tf/Z<.:f: )(f.g>.6 )= )(f,g>. v(a Ma CW: The condition Ce K(f) says that for any two Stolz angles A1 and A2, Cv(d CA1(f,§) = CA2(f,§). So if geK(f), from the preceding paragraph, we have that CA(f,§) = )(f,§) = )(f’C)' Cw(c Cv(d Theorem 21: If f is meromorphic in D and its range of values R(f) is equal to the Frontier of R(f), then, for each g6 C, CA(f,§) = flnA(f,§) A where A varies over all Stolz angles at C. (Rung, 1, Theorem 3, p.49) m: From the hypothesis there exist three values that f assumes at most a finite number of times in D. 30 f is normal by a result of Lehto and Virtanen (l, p.54). Since Interior RA(f,§) = ¢ for any C and any 95 symmetric Stolz angle at g, Theorem 19 implies CA(f,§) = HA(f,C). If A and A are any two Stolz angles at Q, then let A be a symmetric l 2 Stolz angle that contains A1 and A2. Because of the definition of [h(f,§) and CA3(f,§) =HA3(f,§), it follows that CA1(f,§) = HA2(f,§); 3 therefore, XCA(f’C) = 2%(f,§) where A varies over all Stolz angles at g. A function f possesses the p—segment property at g if there exists en: n chords T1,..., Tn at g such that Crk(f,§)fler(f,C) = q, for lgk 1/2, the set of all points C at which f possesses the n-segment property is a set of first category and measure zero on C. (Rung, 1, Theorem 5 , p.50) my Let S(f) denote the set of all g at which f possesses the n— segment property. Then S(f) 0K(f) = o by Theorem 20 since the cluster set along any chord lying in A and terminating at a point of K is CA(f’C)' Since K(f) is a residual set of measure 2TT on C (Theorem 17, Chapter I), S(f) is of first category and measure zero on C. Suppose Y is any boundary arc of D and that 0 < r < 00. Then we define the set J('Y,r) = {zeD :o(z,Y) < r}. If d7 denotes the diameter 0f C ending at T eC, then a boundary are y = z(t) at 1' approaches 1 _ip é m—tangential manner whenever there exists some 0 5‘ to < l and some 0 < r < 0° such that z(t) eJ(d.,,r), t >/ to. The set of all non-tangential boundary arcs at T will be denoted by A(T). Finally we define the sets 96 = = * flu YenA(T)CY(f’T) and nJ(Y,r)(f’T) ?*CY* (f,T) where Y ranges over all boundary arcs at 1' that lie in J(Y,r). Theorem 23 states that for any normal function f in D, any Y, Y' e I = = A(f) and any r, r > 0, nJ(Y,r)(f’T) HJ(Y',r')(f’T) Hu(f,'r). In order to prove this theorem we will need the following two lemmas. m 1: Suppose f is a normal function in D and YE A(T). Let B(z,a) = {zeD :O(z,z') < a] and Zf(w,a) = LZJ'B(z',a) where the union is taken over all 2' 6D such that f(z') = w. If we CY(f,‘r) and there exists an a > 0 such that YflZf(w,a) = (b, then wecy.(f,r) for any Y' EA(T). (Lappan and Rung, 1, Lame l, p.257) Proof: Since we CY(f,'r), there exists a sequence {Zn} on Y such that = §+Zn zn+T and f(zn)+ w as n+°°. Let Sn(C) 1+Cin for ICI < l and f(Sn(§)) = gn(§) for any positive integer n. Because of the normalcy of f, there exists a convergent subsequence {gnk(§)]. If g(g) denotes ' ' ' = lim = lim f(z )= . the limit function, then g(O) k+oognk(0) [(+00 “k w, but, for I“ < tanha , the equation 8nk(§) = w is not satisfied for any value of k. So by Hurwitz's Theorem (Caratheodory, 2, p.195) g(g) 5w. So for 00 any fixed 0 < a' < 00, f +w as z+1 on the set UlB(znk,a'). If Y' e k: MT), then Y' flB(an,a') 7‘ 4) for suitable values of a' > 0 and any posi- tive integer k. Therefore, we CY.(f,T) - Lelnma A: Let f be normal in D. Suppose for some 16 C and for every positive integer n, a set of distinct points {391)}, i = 1,2, ...,mn, with the following properties exists: (1) For some r > 0 and all n, 3;“) eJ(d1,r); 97 (ii) 3§n)+ T as n +00; , (n) (n . = _ . _ (iii) 0(3i ’3i+l) < Kn for i l,2,...,mn 1, With Kn-+~O as n —>ag (iv) there exists a positive number A independent of n such that 0(3in),3rgn)) >,A > 0; n (v) f(3én)) = w for i = l,2,...,mn and n = 1,2,.... Then we5CY(f,1) for all Ye A(f). (Lappan and Rung, 1, Lemma 2, p.258) 2399f: As in the previous lemma, we set f(Sn(C)) = gn(§) for any inte- ger n where now Sn(§) = (g + Ein))/(1 + 3:“)§), Again we denote the convergent subsequence by [gnk(§)} and the limit function by g(g). Since gnk(0) = f(BEnk)) +w as n+°0, g(O) = w. We want to show that the set of points g’ such that g(g') = w which also lie in IQI < tanhzks B is infinite. Suppose there exists a ring R, 0 < r';§ IQ'I,S r" < B with r' r r”, which contains none of the points g'. For any fixed n, the set (Bin): i = l,2,...,mn} is transformed by s;1(z) onto a set of points we call {§§n) :i = 1,2,...,mn] which have the properties: (i') (in) = o and [$12)] >/B; (ii') 0(gén),g§3_i) < Kn for i = l,2,...,mn- 1; (iii') gn(§§n)) = w for i = l,2,...,mn. There must be at most a finite number of gin) for i = l,2,...,mn and any positive integer n within R. Otherwise this set would have a limit point go and by continuous convergence of gn(§) to g(g), g(go)==w, a contradiction of the definition of R. So there exists a positive (:0 integer N such that for n > N no point of the form (1 for i = 1,2,..., mn lies in R. If :11 > N is chosen so that Kn1 < O(O,r") - 0(0,r'), this Violates the properties (i') - (iii') and the definition of R. 98 Consequently g(g) aw in D and the rest of the proof is the same as the last part of Lemma 3. Theorem _2_;: If f(z) is normal in D, Y and Y' are any two arcs in Mr), r.)(f.r) =nu 0, then HJW, (Lappan and Rung, 1, Theorem 1, p.259) r)(f’T) = HJW', 2399:: Using the same notation as in the statement of Lemma 3, we let B(z',a) = {z e D :O(Z,z')4<£fl and Zf(w,a) = %L3(z',a) where the union is taken over all z'esD such that f(z') = w. Then for any fixed curve YEEA(T) and fixed r > 0, let Z%(w,l/n) = Zf(w,l/n)id J(y,r) for n any positive integer. Suppose Y!) ZE(w,l/n) = ¢ for some n. Then the conclusion of this theorem follows immediately from Lemma 3. Now suppose ylW Z%(w,l/n) # o for every n. Then for each n we de— (n)°i=l compose Z%(w,l/n) into its components {Yi . a - "’jn where 1 S j < n\ co}. Assume for each n there exists at least one component YES) whose boundary meets both Y and the boundary of J(Y,r). Then there exists a finite set of points (3?!) :j= l,...,hn} with the properties: (i) 3:") e J(dT,r); (ii) 3ft!) +7 as n+00; (iii) U(3§n),3§:1)< Z/n for j= l, . ..,hn - 1; (iv) 06391)”) < 1/n and U(Eéz), Frontier J(y,r)) <1/n which imply 0(an),3é:)) >/r - 2/n; (V) f(3gn)) = w for j= l,...,hn and n any positive integer. If no is chosen such that 2/no < r/2, then for n >/no the condi- tions of Lemma 4 are satisfied with A: r/2 and Kn: 2/n. 30 the 99 conclusion of this theorem.follows. ( o) n Finally we assume that there exists an nO such that no Yi for i - l,...,jno has a boundary which meets both Y and the boundary of J(Y,r). Let V denote the union of all of the components of Z%(w,l/no) that meet Y and also Y itself. Since this is a connected set lying entirely in J(Y,r), Frlc=={r}. There exists a subset B of Frontier V which is a boundary arc approaching 7 within J(Y,r) and B()Z%(w,l/no) = o. Since WEEC (f,r), this theorem's conclusion follows from Lemma 4. B HOROCYCLIC PROPERTIES OF NORMAL FUNCTIONS In Chapter I we proved properties of horocycles of arbitrary func- tions. In this section we will prove other properties of horocycles which only hold for normal functions. Here we will use some of the same definitions and notations as we used previously. In addition we will use the following definitions. An admissible tangential arc at a point €650 is an arc Y at g for which there exists a sequence {Hr1(n) r2(n) r3(n)(§)] of nested right or nested left horocycles at g with nigmm(r2(n) - rl(n)) = O and each member of the sequence contains some terminal subarc of y. Then Hi%(f,§) = QZCYCf’g) where the intersection is taken over all admis- Sflfle tangential arcs Y at C“ Any point gesc that is both a Plessner point and a horocyclic Ples- sner point of f is called a generalized Plessner point of f. Let Qr(§) denote the interior of the horocycle hr(§). Then the EEAEQEX'tangential cluster set of f at g is defined to be the set = L} . 09039 Or},Tfl { |z| >r}); (iii) i(s,T) = info(so{ Iz|>r},Tn { lzl>r}). r Lemma _6_: If S and T are subsets in D, then B(S) = B(T) if and only if H(S,T) = 0. (Brown and Gauthier, 1, Theorem 1, p.367) Proof: Suppose H(S,T) = O and meB(S). Let {XA} be any net in S that converges to m. We choose YA e T such that 0(xx,y>\) < 20()q\,T) . Since leI +1 and H(S,T) = 0, it follows that 0(xx,y)\) +0. 80 {YA} con- verges to m and 8(8) C B(T). By a similar argument we obtain the inclusion B(T) c 5(3). Therefore, B(s) = B(T). Then we may choose a Blaschke se- zk_—_zn Conversely, suppose H(S, T) > 0. quence {zn } in S such that, for each positive integer n, HIM nz'><5>0 krn and G(zn,T) >/a > 0. From Cima and Colwell (1, p.796) and Kerr- Lawson (2, p.532) it follows that the Blaschke product B associated with the Zn's is bounded away from zero on T. Consequently B(m) 7‘ 0 for each meB(T). Since {Zn} C. S, there is a point meB(S) such that B(m) = 30 6(3) 9‘ B(T). Lemma 1: If S and T are subsets of D, then GflB(S)OB(T) =f o if and only if A(S,T) = 0. (Brown and Gauthier, 1, Theorem 3, p.368) Proof: Suppose )((S,T) = 0. We choose two sequences {zn )l and {2:1} such I I that {zn}eS and {zn]eT, G(zn,zn)< l/n, and kn nllkmzn znzk l/b > 0, n>0. L t o o e m be in B({zn]). We pick a subsequence {2:00} of {Zn} that 104 . l I converges to m . Since n(k)+ww and 0(Zn(k)’zn(A9->O’{zn(l)] converges to m. By Hoffman (l, p.75), m is in G. So Gle(S)(lB(T) # ¢~ _The converse follOWS immediately from Hoffman (l, p.75). Theorem gg: A function f is normal in D if and only if f admits a spherically continuous extension to the set G of regular points of M. (Brown and Gauthier, 1, Theorem 4, p.368) Egpgfz First we will show that if mEEG, then Cf(m) is a singleton Suppose on the contrary there exist two distinct values w1 and w2 in Cf(m) with spherical distance X(wl,w2) = 6 > 0. For each neighborhood V of m, we choose two points 2V and z; in DF1V such that X(f(zv),w1) < 6/3 and X(f(z;),w2) < 6/3. Let S = [2v] and T = {2;}. Then mezB(S)fl B(T)flG and Lemma 7 implies that A(S,T) = 0. By uniform continuity of f, we can pick 216 S and 226 T so that G(zl,zz) < 6 where 6 is chosen so small that X(f(zl),f(zz)) < 6/3. So 6 = X(w1,w2).§ X(w1,f(z)) + X(f(zl),f(22)) + X(f(zz,w2) < 6, a contradiction. Therefore, Cf(m) is a singleton for mesG and we set f(m) = Cf(m). If f is not continuous at m, then for some 6 > 0, each relative neighborhood VFlG of m contains a point mV such that x(f(mv),f(m)) > 6 There exists a ZVE VFID such that x(f(zv),f(mv)) < 6/2. So the net {ZV] converges to m, but X(f(z),f(m)) 2 6/2, a contradiction. Conversely, if f is not normal, then according to Lappan (3, Theorem 1, p.155) there exist two sequences {Zn} and [2;] and an 6 > 0 such that for each n > 0, G(zn,z;)-<>0 but X(f(Zn),f(Z')) > 6- We may Zk-z n l)6 n-lznzk’é assume that the sequence [Zn] satisfies the conditionk so by Hoffman (l, p.75) B({zn})CZG. Since G(zn,Zg)-—>0,1K{Znhiznh = 105 and B({an = 8“th C G by Lemma 6. Suppose m68([zn]). Then for any . g . subnet {zn()\)] converging to m, we also have {znOQ} converging to m. )),f(z' ))) >/e, the cluster set Cf(m) is notasingleton. Since X(f(zn0\ n()\ Theorem 21: If f is a normal meromorphic (holomorphic) function in D and f is the extension of f to the set G of regular points of M, then on each nontrivial Gleason part, f is either meromorphic (holomorphic) or identically equal to infinity. (Brown and Gauthier, 1, Theorem 5, p.369) Mfg: Let me G. Then for any (16 D converging to m, Lu(z) = i converges pointwise to Lm, a one-to-one mapping of D onto P(m) . (Hoffman, l, p.75) We will prove that EoLm is a meromorphic (holomor— phic) function. We pick a fixed point 20 in D and assume that foLm(zo) is finite. Furthermore, we may suppose that (1 lies in some neighborhood of m for which f°L(I is uniformly bounded. For if f°La is not uni- formly bounded in some neighborhood of 20, there exist sequences [Zn] and {on} such that zn—->zO and foLan(zn)+oo . Since f is normal, 1f° Lam} is a normal family of functions. Consequently it contains a subsequence which converges uniformly on compact subsets to a function g meromorphic in D or to 00. Since fOLan+°0 , g(zo) is infinite; however, [f0 La] is uniformly bounded at zo, a contradiction. The family {foLa] converges to fOLm pointwise. Since {fOLa] is uniformly bounded in a neighborhood of z , f0 Lm is holomorphic in a neighborhood of 20. If fo Lm(zo) = co , we look at the family of functions [l/foLa]. The family {foLa} is normal and so it is equicontinuous. Since the Spherical metric is invariant when taking reciprocals, the family of reciprocals {l/foLa} is equicontinuous and thus normal. UfoLm(zo) =0, 106 and,from the previous argument for the finite case, l/f°Lm is holomor- phic in a neighborhood of zo. Therefore, for each point 265D, foLm is either meromorphic (holomorphic) at z or identically infinite in a neighborhood of z. Consequently f°Lm is either meromorphic (holomor- phic) in D or identically infinite. We will now give an example of a normal meromorphic function f such that for each meEM/G, Cf(m) = Rf(m) = W. Let f be a Schwarz triangle function (Carathéodory, 1, Part 7, pp.l73-l94) whose initial triangle is strictly interior to the unit circle. It is well-known that f is a normal function. Let a be any point on W, and let {Zn} be the preimages of a. Since each triangle has the same finite p-diameter, there exists an 6 > 0 such that an 6-neighborhood of [Zn] covers D. By a result of Hoffman (1, Corollary, p.84), B({zn})33 M/G. Therefore, aeERf(m) for each mesM/G. It is an open question whether for each mJEM/G the cluster set is always equal to W. If this is true, then Theorem 27 is also sharp for holomorphic functions. NORMAL HOLOMORPHIC FUNCTIONS In this section we will continue to use the same definitions and notations used in the section of Chapter I which discusses the M— topology for arbitrary functions. Here we will show in Theorem 30 that if f(z) is a normal holomorphic function, then Gf(P) is compact in the M-topology. First of all we will prove in Theorem 28 that any function f(z) which is normal and holomorphic in D belongs to the plgpp :2. This class consists of the holomorphic functions f in D that have the pro- perty that for each pair of arcs t1, t2 6 T(p) along which f(z)—+4» as \ 107 andt . z+p, f(z) is unbounded on each path t between t1 2 Theorem 28: If f is a holomorphic normal function in D, then for each p6C, f is in the class Ip' (Lappan, 11, Theorem 3, p.91) Proof: Suppose pEC and 00 is an asymptotic value of f at p along two disjoint paths t and t in T(p). If t is any path in T(p) between t 1 2 l and t2, then by a remark of Lehto and Virtanen (l, p.53) f(z) +00 as z+p along t. Theorem _22: If p6 C and f6 Ip, then f may have at most two finite asymptotic values at p. (Lappan, 11, Theorem 4, p.91) Proof: Suppose f has three distinct finite asymptotic values a a and ‘ 1’2 a3 at p so that there exist three disjoint arcs t1, t and t3 in T(p) 2 such that f(z) —>ai as z+p along the ti's. Then there exist paths q1 and q2 in T(p) such that qi is between ti and ti+l and f(z) +00 as z+p along qi for i = l, 2. (Remark, MacLane, p.7) So t2 is between q1 and q2 and f is bounded on t2. Therefore, failp Lemma 8: If p 6C and f is a holomorphic function in D which is bounded in a neighborhood of p relative to D, then Gf(p) is compact in the M- t0pology. (Lappan, 11, Theorem 1, p.89) Proof: Suppose Gf(p) is not compact in the M-topology. According to Theorem 35 in Chapter I, there exist directed sequences {tn} and {sn] of arcs in T(p), a number 6 > 0, and a continuum K such that letting 108 Kn=Ctn(f’P) and Ln=Csn(f,p),we have, for any n>0, M(Kn,K) < l/n, d(Ln,K) > 6, and SH is between tn and tn+l' Without loss of generality we may assume that all of the arcs sn and tn originate at the origin, terminate at p a‘nd no pair of arcs have any points in common except 0 and p. Finally we assume M(Kn,K) < 6/2 for all n. Let An be the region It should bounded by t U t + and A' be the region bounded by s U s n n n n l n+1 ° be noted that An and Ah are bounded in the complex plane. Since _ - l l sn {0,p}CAn and tn+1 {0,p]CAn, LnC CAn(f,p) and Kn+1 C CAn(f!P)' According to Collingwood and Lohwater (1, Theorem 5.2.1, p.91), Frontier CAn(f,p) C Ctn(f,p) U Ctn+l(f,p) = Kn U Kn+l Frontier CAI;(f,p) C Csn(f,p) U Csn+1(f,p) = Lr1 U Ln+l‘ Since M(Kk,K) < 6/2 and d(Lk,K) > 6 for every positive integer k, there eXists a point woeLnU Ln+ such that Iwol >sup{|w| :d(w,K) <6/2]. 1 If woeLn, then the fact that Ln is contained in a bounded set whose boundary is KnU K leads to the existence of a point wleKnU Kn+ such n+1 1 that lel > lwol. ‘ However, d(w1,K) < 6/2 violates the choice of WC. If WOELn+1, then there is a similar contradiction. So K6 Gf(p). Lemma 2: Let f be holomorphic in D and p6 C. Suppose further {tn} is a directed sequence of arcs in T(p), K = Ctn(f,p) for n > 0, and K is a n continuum such that M(Kn,K) +0. Then one of the following must hold: (1) Ker(p); (ii) 006 K; (Hi) there exists q between q1 and q3 in T(p) such that f+oo on q1 2 and q3as z+p and f is bounded on q2. (Lappan, 11, Lemma 3, p.90) 109 2399:: Suppose Ké(3f(p), «>6 K and each Kn is bounded. We want to show that (iii) holds. Let Ah be the region bounded by tn U tn+l' If there exists an integer N such that f is bounded in each region An for n > N, then KeaGf(p) because the proof of Lemma 8 only required that f be bounded on a union of three consecutive regions An. Since we are as- suming Ké‘Sf(p), there exist positive integers n1 and n2 with [12 > n1 such that f is unbounded in AHI and Anz. So there exist paths q1 and q3 in T(p) such that q1 —{p} C Ahl, q3 ~{p] C.Ah2, and f(z) —>uaas z-+-p along q1 and q3. Letting q2 = t , we have Cq2(f,p) = an which is [12 bounded. 30 f is bounded on q2 which is between q1 and q3. Theorem 30: If p6 C and fezIp, then Gf(p) is compact in the M—topology. (Lappan, 11, Theorem 5, p.91) E£22£3 If Gf(p) is not compact in the M-topology, then according to Theorem 35 in Chapter I, there exist directed sequences {tn} and {Sn} of arcs in T(p), a number 6 > 0 and a continuum K such that letting Kn CtHCf,p) and Ln = Csn(f,p), we have that for each positive integer n, M(KnaK) < l/n, d(Ln,K) > 6 and sn is between tn and tn+l’ We may assume that “(K ,K) < 6/2 for each n. Since f6 Ip, °<>6 K by Lemma 9. Then n there exists a bounded set L such that Ln.C L for each n and d(L,K) > 6. - I Let A; be the set bounded by Sn U sn+1. f must be unbounded in An for . . 'n each n since Kn+1 C CA$(f,p), Frontier CA$(f,p) CLn U Ln+1 and w is i the same component of the complement of LnLJ Ln+1 as Kn+l' Thus for each n, f has w as an asymptotic value at p along a path qn such that ' r n and f is qn"{P} C Ah' So sn+1 is between qn and qn+1 for eve y b I . ounded on Sn+l' 30 f é p llO Corollary: If f is a normal holomorphic function in D, then Gf is com- pact in the M-topology. This corollary follows immediately from Theorems 28 and 30. NORMAL HARMONIC FUNCTIONS In this paragraph we will show in Theorem 33 that a harmonic normal function has Fatou points on a dense subset of C and in Theorem 34 that a harmonic normal function which does not have +ww asa Fatou value has a set of Fatou points possessing positive measure. Theorem 31: If u is a harmonic normal function in D which omits the value a and if u(z) —>a along a non-tangential boundary path P, then u has a as a Fatou value. (Lappan, 5, Theorem 2, p-154) EEQQfi: Suppose a is finite. Since u omits a, we may assume that u(z) > a for every 2 in D. Let A be an angle containing P and C denote the vertex of A. If {2 ] is a sequence of points in A such that zn-+-§, n . I . there exists a real number M and a sequence of pOints {Zn} in P such . _ l ”I h b- that o(zn,z$) < M. Setting Sn(z) - (Z + Zn)/(1 + an): we ave a 3“ sequence of {u(Sn(z))} converging uniformly in [Z :o(z,0).f M + 1} to a harmonic function U(z). But u(Sn(0)) = u(zg) and so U(O) = a while ”(2) Z.a for zesD. It follows from the minimum principle for harmonic functions that U(z) E a. So u(zn)-+-a and a is a Fatou value of u. SUPPOSG a= +Ww. Defining Sn’ zn and z; as above, we have U(O) = «n HOWeVer, since {u(S (z))} is a normal family, there exists a neighbor- n h00d N of 0 such that u(S (2)) > 0 for n sufficiently large and foN. n 111 It follows from Harnack's Inequality (Ahlfors, 1, Theorem 6, p.183) that U(z) = 00 for ZEN. Consequently U(z) = 00 for ZED. Therefore, u(z)+oo and w is a Fatou value of u. If a = -K5 the argument is similar. Lemma 10: If u is a harmonic normal function in D and v is a harmonic eu(z)+iv(z) conjugate of u, then f(z) = is a holomorphic normal function in D. (Lappan, 7, Lemma 1, p.110) Proof: Let a and b be two complex numbers such that lal # lbl and let [2n] be any sequence of points in D such that f(zn) —>a. Then u(zn) -+ lnlal where 1n0 = «w and lnaa= +om If [2;] is another sequence of points in D such that U(zn,z;) —>0, then by Theorem 4, u(zg)-+-ln}ai since u is normal. Therefore, ’f(z['1)| + 'a| and f(zh) 7913- Using the contrapositive of Lemma 1, we conclude that f is normal. Theorem 32: Let u be a harmonic normal function in D and f = e“(2)+iV(Z? Then every Fatou point of f is a Fatou point of u. (Lappan,7, Theorem 1, p.111) Proof: If g is a Fatou point of f with Fatou value a, then f(z) ~+a and U(z) -+ 1n a as z —+ g from inside each Stolz angle at Q. So C is a Fatou point of u. Theorem 33; The set of Fatou points of a harmonic normal function in D “— is a dense subset of C. (Lappan, 7: Theorem 2’ P'lll) Proof: Let f(z) = eu(z)+iv(z)' Since f is a holomorPhiC normal 112 function, the set of Fatou points of f is dense on C according to Bage— mihl and Seidel (3, Corollary 1, p.16). So by Theorem 32, the set of Fatou points of u is also dense on C. Theorem 24: If u is a harmonic normal function in D such that u does not have +wn as a Fatou value, then the set of Fatou points of u has positive linear measure on C. (Lappan, 7, Theorem 3, p.111) eu(z)+iv(z) Proof: Let f(z) = Since u does not have +w as a Fatou val ue , f does not have m as a Fatou value. Consequently according to Bagemihl and Seidel (3, Theorem 3, p.15) the set of Fatou points of f has positive measure on C. So by Theorem 32, the set of Fatou points of u has positive measure on C. 113 CHAPTER III CLASS A FUNCTIONS INTRODUCTION Let f(z) be holomorphic and non-constant in D. For any complex number a, including w, let Aa denote the set of points C 6 C such that f(z) has the asymptotic value a at C. Let A* = :J Aa and A' = A*lJAw. 300 Then f(z) belongs to Class A if and only if f is holomorphic and non- constant in D and A' is dense on C. Let 8* denote the set of points g 6 C such that there exists an arc F in D ending at C on which If, is bounded on F by some finite constant M. In general M varies as F and g vary. Set B' = B* U A”. Then f(z) belongs to glppg B if and only if f is holomorphic and non- constant in D and B' is dense on G. Since A*<: B* and A' C B', Class A<: Class B. Now let S be any subset of D. For each i>(),0 0, such that (i) D(6) is a conponent of the open set {2: |z| < l, lf(z)- al<6} (ii) 0 < 61 < 62 implies D(el) C D(62) (iii) 0 D(6) = o. 6 > 0 . I I If a = W, the only change in the above definition is to replace f(z)-a < E by lf(z)l > 1/e. Let K = flD(6). Then K is a non-empty, connected closed subset of C and is called the end pf the £5223. If K is an arc, it is called an app- tract. A tract is a global tract if and only if K is the entire Cerum- ference C and for each arc Y(: C there exists a sequence 0f arcs {Yn] SUCh that Yn(: D(l/n) and Yn-a-Y. This last condition is important since Theorem 5 is untrue without some condition of this type 1h the definition of global tracts. 115 If {D(6),a] is a tract and I“: z = p(t), 0 S t <1, is a continuous curve in D such that (p(t) 6 D(6) for l — 0(6) < t < 1, then r belongs to {D(6).a}- Theorem 3: Let f6A and {YD} be a sequence of distinct simple arcs in D which tend to the arc Y of C with the property that zieninlflz)‘ =un+oo as n +00. Then f has {D(6) ,00} with endK such that YCKand for any C 6 K there is a curve T S {D(6) ,00} which ends at Q. At any interior point of K the only asymptotic values come from this tract. (MacLane, 1, Theorem 3, p.15) Theorem 4: Let f6A and let {D(6),a] for a 3* 00 be a tract of f. Then the end of this tract is a single point. (MacLane, 1, Theorem 4, p.18) Theorem 5: Let f6A. Then (i) f has a global tract if and only if f is unbounded and all level curves of f are compact; (ii) f has a global tract if and only if f is unbounded on every curve I‘ in D on which |z| +1. (MacLane, 1, Theorem 6, p.18) Theorem 6: If f6A and S is any Borel set on the sphere, then A(S) iS measurable. (MacLane, 1, Theorem 10, P-Zz) SUFFICIENT CONDITIONS FOR f EA Theorem 7: Each one of the following conditions is a sufficient condi- ‘— tion for f6A: 116 (i) f is a holomorphic, non-constant function in D such that there exists a set Se C [0,2fl] thatis dense in [0,2fi] such that £}(l -r) log+ |f(reie)ldr < M>for 96 Se; (ii) f is a holomorphic, non-constant function in D such that [61(1 - r)m(r)dr < 00 where m(r) = ifjfilogflflreieflde, Ogr< 1; (iii) f is a holomorphic, non-constant function in D such that Jél(l - r) log M(r)dr N. Then fszA. (MacLane, 1, Theorem 16, p.42) f(Z) belongs to the Class N_if and only if f is holomorphic, non—r constant in D and normal. Theorem g: N<: A. Also, if f6 N, then (i) given g6 C, f has at most one asymptotic value at g. If f has the asymptotic value a at Q, then f has the angular limit a at g; (ii) f has no arc-tracts. (MacLane, 1, Theorem 17, p.43) Bagemihl and Seidel results in ChapterII, This theorem contains the Theorems 4, 12 and 14. 117 BARTH'S GENERALIZATIONS OF MACLANE'S RESULTS In order to generalize MacLane's Results, Barth defined classes Am, Bm, LIn and L; which differ from MacLane's classes A, B, L and L* only in the replacement of the word "holomorphic" with the word "meromorphic" in the appropriate definitions. Theorem 12 shows that AmC Bm and Lm C Bm. However, there are examples to show that no other inclusion relation— ships exist among the classes Am, Bm and Lm' Let LSOx): 126D: |f(Z)| =A1- Theorem m: Let f6Am and {Yn] be a sequence of disjoint simple arcs in D that tend to the arc Y of C with the property that there exists a com— plex number a such that sup|f(z) - al = pn+0 as n+°° if a =r‘ °°, Yn inf|f(z)|= Mn —+oo as n + 00 if a - cc. Yn Then f has an arc tract {D(6),a} with end K such that Y C K and such that for each point QEK some curve I‘ belonging to [D(6),a} ends at g. At any interior point of K, the only asymptotic values come from this tract. If f6L , the preceding conclusions are true for a = 00. m (Barth, 1, Theorem 1, p.323) _ 19. Proof: First we assume that f6Lm and a = 00. Let Y " {e ' <15 9 5 B] I l and g be an interior point of Y. We choose CLUB Sheh that a < a < argg < B' < B. Let S(a',B') denote the sector {(1' < argz , N(l) no Yr1 intersects LS()\). Let the components of LS()() 0 En be denoted by p(n,i), for l\< i \< ni. For sim- plicity we pick 1 so that LS(1) has no multiple points. Since feLm, the maximum diameter p(n,i) for n 3 N()\) approaches zero. So for n>/N1, any curve p(n,i) which intersects the radius R = [arg z= argg] is a Jordan curve contained in En' Therefore, any interval of R in En on which If(z)| < A may be replaced by an arc of a level curve p(n,i). By making a finite number of such replacements for any one [1, we obtain a curve TO.) such that z 131 zi€h£(x)lf(z)l >/)\- We will now construct I‘. Let AnT 00 be given. Let Qn be the in- tersection of R with Yh having max Izl, and let F(lkm) be the portion of PO‘k) joining Qn to Q. We define I‘= F(kl) from z = O to in where n1 is chosen so that largz- arg§| < 1/2 and lf(Z)l >/)\2 for ZEI‘(A2,H1); to Qn where n > n is 1 p p-1 chosen so that Iargz- arggl < l/2p and lf(z)l;1k for Z6 F(Xp,np_1). and for any integer p > 1, F= P(lp) from Qn p- So P+C and f(z) +00 on 1‘. Now we assume that g is an endpoint of Y. Let {gm} be a sequence 0f interior points of Y with §n+§ and I‘n be a curve ending at Cn on which f +00. By using a construction similar to the one given above, We can construct a curve I‘ tending to Q on which f +00. Each asymptotic path I‘ to an interior point g of Y intersects all Y 's for n > N. So there exists an integer N' such that all Yn for h> N' belong to the same domain D(6) for If(z)l > l/E. Thus all Paths belong to the same tract [D(6),°°}. If the end K of this tract 119 contains Y as a proper subset, then we can choose arcs Y; CID(1/n) such that Yé-a-Y' = K. Since Y; and Y' satisfy the same hypotheses as Yn and Y, it follows that if CE K, then there exists a curve I belonging to [D(6),m} which tends to g. If fesAm and a = m, then LS(1) fl S(a,B) must also end at points of C for all I > 0. For if this were not true, there would exist a 11 > 0, a subarc A of Y and a sequence of continuous arcs {Ah} compact in D such that An C LS()\1) for all n and An +A as n+°°. Let Q be any interior point of A. Each curve ending at C must cross all but a finite numberof the Aq£s and‘Yn's. Therefore, f cannot have an asymptotic value at g, contradicting the assumption fesAm. Finally, if a is finite, we define the function l/(f-—a) and use the above proofs. Ihgggem 11: If fesLm and Y = [e16 :q.S 0 S B,a #8) is a subarc of C such that no level curve of f ends at any point of Y, then exactly one Of the following two statements is valid. (1) For each interior point em (a < ¢ < B) of Y, there exists a ' i . continuous curve f(e1¢) C1D ending at e ¢ and such that f is bounded on L} f(ei¢). Furthermore, f does not have m as d<¢/N. (iii) there exists a TnC Boundary A(An) such that Tn is a cross- cut of the sector S(a,B) that joins a point of arg z =(1 to a point of argz =B. (iv) Boundary A(An) 3 Y. If (iii) is true for all n >/N, then Tn+Y and so by Theorem 10, f has an arc tract {D(6),oo} with end KC Y. So (ii) holds. Now suppose there exists some n = M for which (iv) holds. Let Q = ei‘D for a < 4) < B be any arbitrary point of Y. By (iv) CE Boundary A(AM). Since f6 Lm and no level curves of f end at points of Y, there exists a 6 > 0 depending on g such that each component of Boundary A(XM) having non-empty intersection with the set [z :Iz— d < 6, |z| < 1} is a closed Jordan curve contained in S((I,B). Since the diameter of the set LSQM) fl [2:1-6 < |z|< 1} +0 as 6+0, 0 and g may be connected by a continuous curve New) C A(AM) U C' The last part of (i) is proved by observing that the existence of the asymptotic value 00 at g implies that L80.) ends at g for all >‘>>‘M’ which is a contradiction. Theorem 12: A C B and L C B . (Barth, 1, Theorem 3: P325) — m m m m k 121 Proof: Since the generalized definitions and notations include "mero— morphic functions" instead of only "holomorphic functions", A'= A*UA°O and B = B’" U Am. So A*C B* and A' C B'. Conse- where A*= U A a7‘oo a A C . quently m Bm i6 t C . = : 1 Now we wan to Show that LIn Bm Let f6 Lm and Y {e (is 9°] with end K con- taining Y. In the first case we are finished; in the second case by Theorem 10, f has the asymptotic value 00 at each point of Y. The Schwarz triangle function is an example of a function f such that f6 Bm and f6 Lm’ but féAm. This shows that Bm (Z Am and LmC Am. We will now construct a function f such that fEAm, f6 Bm, but féL . (Barth, Example 2, p.326) Let [rm] denote a sequence of posi- m tive numbers which are strictly increasing to 1. For n >, 1, let C = {IZI = rn] D = {|z| < rn] n n — = =o,1,...,2 -1. E - {z : rn SIZ|S rn+1 and argz 2kTT/2} for k For n > 1 let F = D U E U C . Two sequences of functions ’ n n n-l n -l [fn(z)1 and {R (2)] are now defined inductively. n W (D _ = E . N t We define f1(z) and R1(z) on Dl so that f1 _. R1(z) 1/2 ex 122 construct f2(z) so that it is continuous on F and 2 f2(z) = f1(z) on D1, f2(z) = 5/4 on C2, f2(z) is linear on each component of E1. F2 is closed and it divides the plane into a finite number of regions. In addition f2(z) is continuous on F2 and analytic on the interior of F2. Therefore, by a remark in Mergelyan's paper (1, p.24) there exists a rational function R (2) such that max If (z)-R (2)1 < 2-4. In gen- 2 ze F2 2 2 eral suppose that fn(z) is spherically continuous on Fn and that fn(z) = Rn-1(Z) on Dn-l’ n -n = + _ fn(z) l ( 1) 2 on Cn’ fn(z) is linear on each component of En-l' By using a remark of Mergelyan (l, p.24) we can find a rational function -n-2. A straightforward calcula- Rn(z) such that 21?; Ifn(z)-Rn(z)! < 2 tion shows that [Rn(:)] converges to a meromorphic function R(z) in D. In order to show that R(z) élhfi it is sufficient to prove that for each n some component of [z :IRI = 1} separates Cn and Cn+l' This is shown by verifying that IR(z)- (li-(—1)n2-n)| < 2‘“.1 for Z6 Cn' Fur— thermore, f ( the limit of [fn(z)}) has the asymptotic value 1 on each radius of the form [2 :0.S |z| < l and arg z=2_nk) for n > 0 and k = 0, 1,...,2n_1. Since these radii are dense, fezAm, f6 Bm and f6 Lm. Barth has established some sufficient conditions for a function to be a member of A . Theorem 14 shows that the conditions 0f Theorem 7 m can be generalized to meromorphic functions. 123 Theorem 13: Let g and h be holomorphic in D and let g/h be nonconstant. Suppose g6A, h is bounded and f = g/h. Then f6Am and l/feAm. (Barth, Theorem 6, p.331) M: Let Y be any subarc of C. We will show that there exists a point §6Y and a curve ending at g on which f tends to a limit as [Z] +1. First suppose Aw(g)fl Y 9‘ (1). Then there exist a point §6Y and a curve I‘ ending at g on which g +00 as 12' +1. Consequently, since h is bounded, f +00 as [Z] +1 on I‘ and f has the asymptotic value 00 at g. Now suppose A00(g)fl Y = ¢~ If g is bounded in some neighborhood of a point g on Y, then f has an asymptotic value at g by the Fatou Theo- rem (Fatou, 1). So suppose zlgngsuplgw)! = °° for all Q6Y. Under these hypotheses MacLane (l, p.26) has shown that there exists a AC D with the following properties: (i) A is a simply connected Jordan domain, bounded by crosscuts 1‘ of D on which lgl = 1 for some )\ > 0 and by a nonempty subset F of y. (ii) lg(z)l < N whereNis apositive integer for all 26A. (iii) There exists a nonempty subdomain A' of A such that 1 < l g(z)] < N for all Z6A'. Based on an argument of MacLane (1, p.27) for the proof of Theorem 2 of this chapter, it can be shown that f has asymptotic values at some points of Y. Consequently f6Am and l/f is also in Am. We are now ready to generalize the conditions (i), (ii) and (iii) . ' 'ons in Theorem 7 to obtain sufficient conditions for meromorphic functl n—r—w 11 124 to be in Am' (Barth, l, p.332) Let f be meromorphic in D. Condition (i') Suppose there exists a complex number a, possibly w,and a set 0 dense on [0,2fi] such that the Nevan- linna counting function N(r,a) = 0(1) (Nevanlinna, 1) and fl(l -r) log+ 1 dr < m for eese if 0 19 f(re ) -a area. If a=oo, then Ilsa-r) log+lf(rele)‘drwhere T(r) is the Nevanlinna characteristic of f. l f(rele)-a l + . Since Condition (ii') implies that £)(l—r)log dr<2and1‘1=Y1Uu11 127 This function f(z), which Barth and Schneider construct, is not in A. It is bounded away from zero on a countable set of asymptotic paths {an} which are tangent to those radii ending at a countable dense subset {tn} of C. According to Privalow (p.214), there exist nonzero functions analytic and bounded in D which have radial, and hence angular, limit zero on any pre—assigned subset N of C of measure zero. Let h(z) be the particular function obtained when N = Eatn. h(z) = w(z)/b(z) is in n- Class A since 1im h(z)=(n for each on while w(z)==h(z)'b(z) 6 A. Z6011 1z 1m>l Recently Tse (1) has shown a condition which holds whenever a pro- duct of a bounded holomorphic function and a Class A function is not in Class A. If f(z) is a meromorphic function in D, then we define Ff(K) [or F¥(K)] for 0 s K g x>to be the set of Fatou points of f(z) on C at which the Fatou values are greater than [or less than] K in absolute value. Theorem 15: If b(z) is a bounded holomorphic function in D and if f(z)6.A, but f(z)b(z)é.A, then Fb(0) is of first category in some subarc of C. (Tse, 1, Theorem, p.68) Proof: Let Aw(fb) denote the set of points g for which (6 C and fb has w as its asymptotic value. Let B*(fb) denote the set of points g such that g6 C and there exists an are P in D ending at C on which lfl is bounded by some finite constant. So be = B*(fb) U Am(fb). Since f(Z)b(z)¢_A= B, there exists a subarc Y of C such that be 0 Y = ¢. By definition Fb(0) H Y = ngFb(l/n) 0 Y. We will show that Fb(0) 0 Y is of first category. Suppose on the contrary it is of second category. 128 Then there exists an no > 0 such that Fb(1/n) n Y is of second category. So at each point g6 Fb(l/no)FIY, the radial cluster set of b(z) does not contain the value 0. By Collingwood (2, Lemma 1) there exists a number M' > 0 such that 1l/b(z)1 S M' in a neighborhood U of a subarc B of Y. Therefore, 0<1/M'g|b(z)|gMat points of Y. Let g be any one of these points in the interior of Y. We choose a tract T(6) so that 1f1 > 1/6 near g and T(6) (I C C Y. We also pick ZOET(E) and consider the Riemann surface over the w-plane corresponding to T(6). Thereisea6, 0 < é<1T/4, such that sector {w : —6 jta> on the preimage F of this ray. So ef+0 orooon F. CALCULUS PROPERTIES OF CLASS A FUNCTIONS MacLane (2) and Barth and Schneider (2) have investigated the ques- tion, "If fesA, then what are sufficient conditions for f'EEA or £ff(£)d£ 6 A‘W' The latter have also studied similar conditions for functions in Class Am. Let J denote any domain bounded by a Jordan curve K and lying in G. Then A[J] is the set of nonconstant functions f holomorphic in J with asymptotic values at every point of a set of points S}\ 3‘ 00 as u f 1, (iii) either (a) I‘ is rectifiable or (b) ¢1(w(u))=¢(¢(u)) -q(w(u)) is of bounded variation on [0,1], then f has a finite asymptotic value on I‘. (MacLane, 2, Lemma, p.273) Proof: Consider the differential equation < > “'1 (m) w n(z)+ z p (z)w (z)= ¢(z) -q(z) for zeA' (2) m=o m where q; is the function defined in (ii). Then f(z) is a solution in A'. Let A* denote the component of A' which contains I‘~¢(1) where w(l) is understood to be X. Let g1(z), ..., gn(z) be a set of linearly indepen- dent solutions of the homogeneous differential equation associated with Equation (2). The functions gi are holomorphic in A0. By variation of parameters the solution of Equation (2) is given by n z = + - * f(z) Zl{a J; hm(t)[¢(t) q(t)] dt} gm(z) for zeA where a=\1;(0) and (1m are constants. hm are functions holomorphic in A0. 132 In Case (iii) (a), the integrand is continuous on F and Bm = J": hm(t) [¢(t) - q(t)] dt = I: hm(t) ¢1(t) dt has the finite asymptotic value 1 Bml = $0 th (u)) [cm (u)) ~q - ¢1(a)Hm(a) - fa Hm(t)d¢1(t>. Each of the first two terms has a finite asymptotic value on F. Also ffiumd¢l = Jmew>d¢l<¢> ->folnm(¢(u)>d¢l<¢(u>> because Hm(w(u)) is continuous on [0,1] and ¢1(w(u)) is continuous and of bounded variation on [0,1]. Theorem 12; Suppose f(z) is holomorphic and nonconstant in D. Let A* = {lz-ll < r}, n be a positive integer and po(z), ..., pn-1(z), q(z) be holomorphic functions in.Afly Let A1= ner? D and n-1 ¢(z) = f(n)(z) + 2 p (z)f(m)(z) + q(z) for 2623. m=o m If ¢EEALA] and there exists a finite constant c such that ¢1(z) = ¢(z) - q(z) # c, then fesA[AJ and A*(f), the set of finite asymptotic values, is dense on car? C. (MacLane, 2, Theorem 1, p.275) Proof: Sincezfi* can be replaced by a smaller disk contained in A? with its center on C, it is sufficient to show that f possesses a finite asymptotic value at one point on Afiflc. 133 Let E denote the subset of points g on C such that for each point g there exists a neighborhood UQ = [iz-§| < r}r) D and a Jordan arc JC such that ¢(z) maps UQ into the complement of JC. In a similar manner E1 is defined using ¢1 = ¢(t)-—q(t) instead of ¢. We set E2 = E LJEl. Suppose AfiWW C contains points of E2. By shrinking.A* we may as- sume.&*f) C is contained in E2. From a simple generalization of Fatou (1) both ¢ and ¢1 have finite angular limits almost everywhere since u(z) - ¢1(z) = q(z) has angular limits almost everywhere. Using the notation in the proof of Lemma 1, we see that 8m has finite angular limits almost everywhere. So f also has finite angular limits almost everywhere on.A*F\C. From a theorem of Privalow (l, p.210) the asymp- totic values assumed by f(z) on any interval Afllfi C contained in E2 form a set containing a closed set of positive harmonic measure. Consequently this set must be infinite. If E2 is dense on.A*rW C, we are finished. So we now assume that.A*{) C is contained in the complement of E2. According to MacLane (1, Theorem 7, p.19) each asymptotic tract of ¢1 must end at a single point because ¢1 omits the value c. Suppose that the asymptotic values of ¢1 are bounded by a finite constant M. We choose two distinct points g1 and g2 on.A*() C at which ¢1 has asymptotic values and join C1 and g2 by a curve F(Z‘A which is an asymptotic path at both Cl and g2. Then ¢1 is bounded on T and we denote the bound by B. Let G be the domain bounded by F and part of Afif) C. If ¢1 is bounded in G, then the arc §1§2 is contained in E2, a contradiction. So we pick a value w such that w = ¢ (z ) for some 2 EEG and [W I > o o 1 o o o max(B,M). Then by the lifting argument of MacLane (1, p.13, Section 2), ¢1 has an asymptotic value a at some boundary point of G on C satisfying the condition x);.a >"wo| > M,eacontradiction. Therefore, we now assume 134 that ¢1 has two asymptotic values along r, whose magnitudes are greater than 2IcI, where c is the constant defined above. So 1¢1-c‘ Z 6 > 0 on r. Since ¢1 omits fewer values in G than a Jordan are, there exists a 216 G such that |¢1(zl) - cl < 6. By the lifting argument of MacLane (l, p.13, Section 2), there exists r1(: G ending at a point of A*f) C such that ¢1(z) maps f1 one-to-one onto a linear segment. By Case (iii) (b) of Lemma 1, f has a finite asymptotic value on F1. Special cases of Theorem 19 show that for any positive integer n if f(n) EA and f(n) 9‘ c, then feA and A*(f) is dense on C. Theorem 29; Let fezA and f(z) # c, where c is some finite constant. Then A: is dense on C. (MacLane, 2, Theorem 5, p.278) 2529:: We may assume without loss of generality that c==0. Let Y be an arbitrary arc of C. First we suppose that there is an interior point of Y such that Tim inflf(z)l = O. From MacLane (1, Theorem 11 and its 2 Corollary, pp.25 - g8) we can find a crosscut F of C from gle‘Y to gze‘y with the properties that f has nonzero asymptotic values on F at both g1 and Q2 and Q is in the open are from £1 to C2. Let A be the domain bounded by r and the open are from £1 to g2. |f(z)l 2 m > 0 on F. We pick a point zo 0 for all Za and gn-—>g. Another result is that if f'eaA and f'(z) # 0, then f possesses at least three numerically distinct asymptotic values since f‘sA by Theorem 19. For n 3 1, any function f meromorphic in D and any zeED, we define the ”nth integral of f" as ('11) _ 2 $1 $2 Err-l F (z) —J; J; J; ...J; f(gn)dsndgn_l...dsl where the Ei's are dummy variables. In order to eliminatethestatement, 137 "If f'(z) is nonconstant," in our theorems, we define the Class A? to be the union of all functions in Class A and the constant functions. Theorem 21: If f is holomorphic in D and satisfies the integral part of Condition (ii) of Theorem 7, then f'esA*. (Earth and Schneider, 2, Theorem 1, p.4) Proof: First suppose f(0) = a # 0. Using the notation 2W . m(r,f) = 2%“; 10g+|f(rele)ld9, we have m(r,f') = m(r,f'(f'/f)) S m(r,f) + m(r,f'/f). According to the ”logarithmic derivative lemma" of Nevanlinna theory (Hayman, l, p.36) + m(r, fo) <4log+m(R, f) + 410g+(10g+l—f_(16)—]) + 510g R+ 6log+R-}}+ log+ %+14 where 0 < r < R < 1. Suppose r > 1/2 and let R = (r+l)/2. Then m(r,f'/f) < 4 log+un((r+l)/2,f) + 6 log+-(2/(l-r)) + K where K is a constant that depends on a, but not on f. Therefore, 1 1 1 fen—nmnfvdr g o(l-r)m(r,f)dr+ J;(1-r)[6logf—r+ KJdr 1 + 4J;2[1 - (#1)/2]1og+m(.g(r+1),f)dr. Because of the hypotheses of this theorem, all the integrals on the right hand side of the last inequality are finite. Consequently f' satisfies the integral part of Condition (ii) of Theorem 7. So f'esA*. Actually the previous proof demonstrates that if f and f' are 138 nonconstant and f satisfies Condition (ii) of Theorem 7, then f' satis- fies it also. So we have the following Corollary. If f is holomorphic in D and satisfies Condition (ii) of Corollary 3: Theorem 7, then f(n)(z) eA* for all n 2 0. (Barth and Schneider, 2, Corollary 1, p.6) If f is holomorphic and normal in D, then f(n)(z) eA* for Corollary 11;: n 2 0. (Barth and Schneider, 2, Corollary 2, p.6) Proof: According to MacLane (1, p.44) if f is holomorphic and normal in D, then m(r,f) g Cllog(l/(1—r)) + C2 where C1 and C2 are constants. Hence f satisfies Condition (ii) of Theorem 7. Theorem g: If f is holomorphic in D and satisfies (l-r) log(l/(l-r))m(r,f)dr < 00, then f and F(-1)(z) e A". (Earth and Schneider, 2, Theorem 2, p.7) M: Since the integral condition in this theorem's hypothesis is stronger than the integral part of Theorem 7's Condition (ii), feA*. In order to establish that F(_1)(z) e A*, we will use the theorem of Hayman (2, Theorem 2) which states that if F(z) is holomorphic in IzI s (1+ (1/11))10g(4/(1 — r))m((r+ 1)/2,f). Consequently I (_1) l ,% (l-r)m(r,F )drgf‘g (l-r)m((ri'l)/2,f)dr 1 1 z, + fi 02[l—(r+l)/2](logifi:?;;is7ijhn«r+l)/2,f)dr. Because of the hypotheses of this theorem, both of the above integrals . -1) of the right hand side of the inequality are finite. Consequently F( . . (-1) 1 satisfies the integral part of Condition (11) of Theorem 7. So F GA}. Theorem 21 and 22 may be generalized to meromorphic functions f for which the Nevanlinna counting function N(r,f) = 0(1). Let A; denote the union of the functions of Class Am and the constant functions. 1 ihggrgm 23: If f is meromorphic in D, N(r,f)==0(l), and‘£(l-r)T(nf)dr 7': < m, where T(r,f) is the Nevanlinna characteristic of f, then f'eEAm. (Barth and Schneider, 2, Theorem 4, p.11) Since T(r,f) = m(r,f) + 0(1) and T(r,f') = m(r,f') + 0(1), this Proof is completely analogous to that of Theorem 21. Theorem 24: If f and F(-1) are meromorphic in D, N(r,f) = 0(1) and fl(1-r)log(l/(l-r))T(r,f)dr < oo, 0 then f and F—1(z) e A*. (Barth and Schneider, 2, Theorem 5, p-lz) m \ 140 The proof of this theorem is quite similar to that of Theorem 22 where the meromorphic form of Hayman's Theorem (2, Theorem 2) replaces the holomorphic one. BIBLIOGRAPHY Ahlfors, L. V. (l) Zur Theorie der Uberlagerungsflachen, Acta Math. 65, 157-194 (1935). —.—__—_—.—__——_—_——_—_ Proc. London Math. Soc. (3) 14, 260-270 (1964). Bagemihl, F. (l) Curvilinear cluster sets pf arbitrary functions, Proc. Nat. Acad. Sci. 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(3) Uber die gegenseitige Beziehung der Rander bei der Abbildung des Innern einer Jordanschen Kurve auf einen Eppip, Math. Ann. 73, 305—320 (1913). (4) Uber die Begrenzung einfachzusammenhangender Gebiete, Math. Ann. 73, 323-370 (1913). Carleson, L. (1) Interpolation py_bounded analytic functions and the corona problem, Ann. of Math. (2) 76, 547-559 (1962). 143 Church, P. T. (1) Ambiguous points pf a function homeomorphic inside a 3 here, Mich. Math. J. 4, 155-156 (1957). Cima, J. A. (1) A_nonnormal Blaschke-quotient, Pac. J. Math. 15, 767-773 (1965). Cima, J. A. and P. Colwell. (l) Blaschke quotients and normality, Proc. Amer. Math. Soc. 19, 796-798 (1968). Clunie, J. (1) 03 p problem pf Gauthier, Mathematika 18, 126-129 (1971). Collingwood, E. F. (1) 9p the linear and angular cluster sets pf functions meromorphic pp the unit circle, Acta Math. 1, 165-185 (1954). (2) 93 sets pf maximum indetermination g: analytic functions, Math. Z. 67, 377-396 (1957). (3) Cluster sets pf arbitrary functions, Proc. Nat. Acad. Sci. U. S. A. 46, 1236-1242 (1960). (4) Cluster set theorems for arbitrary functions with applications pp function theory, Ann. Acad. Sci. Fenn. Ser. AI 336/8, 1-15 (1963). Collingwood, E. F. and M. L. Cartwright. (1) Boundary theorems for 3 function meromorphic ip the unit circle, Acta Math. 87, 83-146 (1952). Collingwood, E. F. and A. J. Lohwater. (1) The theory pf cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, Cambridge, 1966. Dolzhenko, E. P. (1) Boundapy properties pf arbitrary functions (in Russian), Izvectya, Akad. Nauk SSSR 31, 3-14 (1967). English translation: Math. of the USSR-Izvestija 1, 1-12 (1967). Dragosh, S. (1) Horocyclic boundapy behavior pf meromorphic functions, J. d'Analyse Math. 22, 37-48 (1969). _—————————_—_—————_ disc, Nagoya Math. J. 35, 53-82 (1969). Fatou, P. (1) Series trigonometriques pp series q; Taylor, Acta Math. 30, 335-400 (1906). 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(1) Meromorphic Functions, Oxford University Press, London, England, 1964. ____.—__—_—_—_—.___ 214 (1964). Hille, E. (1) Analytic function theory, vol. II, Ginn and Company, New York City, New York, 1962. _._—____—————.___— third ed., Harren Press, WaShington, D. C. 1950. Hoffman, K. (1) Banach spaces 9f analytic functions, Prentice-Hall, Englewood Cliffs, N. J., 1962. (2) Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86, 74-111 (1967). Hunter, U. (1) An abstract formulation of some theorems on cluster ——_———-—————._——_—.-——-———__ sets, Proc. Amer. Math. Soc. 16, 909-912 (1965). (2) Essential cluster sets, Trans. Amer. Math. Soc. 119, 380- 388 (1965). Kaczynski, T. J. (1) Boundary functions for functions defined in a disk, J. Math. Mech. 14, no. 4, 589-612 (1965). 145 (2) 9p p boundary property pp continuous functions, Michigan Math. J. 13, 313-320 (1966). Kerr-Lawson, A. (1) A filter description pp the homomorphisms _f °", Canad. J. Math. 17, 734—757 (1965). (2) Some lemmas pp interpolating Blaschke products and p cor- rection, Canad. J. Math. 21, 531-534 (1969). Krishnamoorthy, S. (1) Boundary properties p: pp infinite product Z. 114, 93-100 (1970). Lappan, P. (1) Non-normal sums and products p: unbounded normal functions, Michigan Math. J. 8, 187-192 (1961). (2) Sums of normal functions and Fatou points, Michigan Math. J. 10, 221-224 (1963). (3) Some sequential properties p£_normal and non-normal functions with applications pp automorphic functions, Comm. Math. Univ. Sancti Pauli 12, 41-57 (1964). (4) Some results on harmonic normal functions, Math. Z. 90, 155-159 (1965). (5) Asymptotic values pi normal harmonic functions, Math. Z. 94, 152-156 (1966). (6) Some results pp functions holomorphic in he unit disk, —._~ Comm. Math. Helv. 41, 183-186 (1966-67 . (7) Fatou points pg harmonic normal functions and uniformly normal functions, Math. Z. 102, 110-114 (1967). (8) A property pp angular cluster sets, Proc. Amer. Math. Soc. 19, 1060-1062 (1968). (9) Continua which are curvilinear cluster sets, Nagoya Math. ————————_—___——__ J. 34, 25-34 (1969). (10) A characterization p: Plessner points, Bull. London Math. Soc. 2, 60-62 (1970). (11) Arc cluster sets pf_holomorphic functions, Yokohama Math. J. 18, 87-92 (1970). Math. Univ. Sancti Pauli 18, 119-124 (1970). (12) Some results on a class of holomorphic functions, Comm. (13) A note _p_p problem pf_Gauthier, Mathematika (Dec., 1972). 146 Lehto, O. and K. I. Virtanen. (1) Boundary behavior and normal meromorphic functions, Acta Math. 97, 47-65 (1957). Proc. London Math. Soc. (2) 28, 383-394 (1928). Lohwater, A. J. and G. Piranian. (l) The boundary behavior pi func- tions analytic ip p disk, Ann. Acad. Sci. Fenn. Ser. AI, no. 239, 1-17 (1957). MacLane, G. R. (1) Asymptotic values pi holomorphic functions, Rice Univ. Studies 49, no. 1, 1-83 (1963). (2) Exceptional values pi f(n)(z), asymptotic values pi f(z! ppp linearly accessible asypptotic values, Math essays dedicated to A. J. Maclntyre, Ohio Univ. Press, Athens, Ohio, 271-288 (1970). Mathews, H. T. (1) A note pp Bagemihl's ambiguous point theorem, Math. Z. 90, 138-139 (1965). ———_—.———— Mathews, J. H. (l) Asypptotic behavior pi light interior functions defined in the unit disk, Amer. Math. Soc. Proc. 24, 79- 81 (1970). (2) Asymptotic values pi normal light interior functions defined ip the unit disk, Amer. Math. Soc. Proc. 24, 691-695 (1970). McMillan, J. E. (1) Boundary properties pi functions continuous ip p disc, Michigan Math. J. 13, 299-312 (1966). (2) A boundary property pi holomorphic functions, Math. Ann. 173, 275—280 (1967). (3) Boundary behavior pi conformal mapping, Acta Math. 123, 43-67 (1969). McMillan, J. E. and Ch. Pommerenke. (l) Qp the asymptotic values pi locally univalent meromorphic functions, J. Reine Angew. Math. 249, 31-33 (1971). (2) 9p the boundary behavior pi analytic functions without Koebe arcs, Math. Ann. 189, 275-279 (1970). Meek, J. (1) Subharmonic versions pi Fatou's Theorem, Proc. Amer. Math. Soc. 30, 313-317 (1971). 147 A com- ) 7, no. 2 (48), 31-122 (1952). Amer. Math. Soc. Transl. no. 101, Providence, Phode Island, 1954. Mergelyan, S. N. (1) Uniform approximations ip functions pi plex variable (in Russian), Uspehi Mat. Nauk (N. S. Nagatomo, J. (1) Op p problem pi MacLane, Proc. Japan Acad. 44, 879-883 (1968). Nevanlinna, R. (1) Eindeutige analytische Funktionen, 2te Auf1., Springer-Verlag, Berlin-Gottingen-Heidelberg, 1953. Newman, M. H. A. (1) Elements pi the topology _i plane sets _i points, Cambridge University Press, 1961. Noshiro, K. (1) Cluster sets, Springer—Verlag, Berlin, 1960. (2) Contributions pp the theory pi_meromorphic functions ip the unit circle, J. Fac. Sci. Hokkaido Univ. 7, 149-159 (1938). Painleve, P. (l) Lecons sur ip theorie analytique des equations differentielles professees p Stockholm 1895, Hermann, Paris, 1897. (2) Sur les singularites des fonctions pi, pp particulier, des fonctions definies par les equations differentielles, C. R. Acad. Sci. Paris 131, 489-492 (1900). Pommerenke, Ch. (1) Normal functions, Mathematics Research Center, Naval Research Laboratory, Washington, D. C., Proceedings of the NRL Conference on Classical Function Theory, 1970. (2) 9p Bloch functions, J. London Math. Soc. (2), 2, 689-695 (1970). Privalow, I. (l) Randeigenschaften analytischer Funktionen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1956. Rung, D. C. (1) Boundary behavior pi normal functions defined ip the unit disk, Michigan Math. J. 10, 43-51 (1963). Ryan, F. and K. Barth. (l) Asymptotic values pi functions holomorphic ip the unit diSk, Math. Z. 100, 414-415 (1967). Schneider, W. (1) Ap elementary proof pi p theorem pi MacLane, Monatsh. Math. 72, 144-146 (1968). Seidel, W. and J. L. Walsh. (1) Op the derivatives pi functions analytic ip the unit circle and their radii pi univalence and p-valence, Trans. Amer. Math. Soc. 52, 128-216 (1942). Stebbins, J. (1) A note pp extended ambiguous points, Nagoya Math. 148 1 J. 43, 167-168 (1971). Titchmarsh, E. (l) The theory pi functions, 2nd ed., Oxford Univ. Press, London, England, 1939. Tse, K. F. (1) Op p theorem pi Nagatomo, Yokohama Math. J. 18, 67-69 (1970). Tsuji, M. (1) Potential theopy ip_modern function theory, Tokyo, 1959. Vessey, T. A. (1) Tangential boundary behavior pi arbitrary functions, Math. Z. 113, 113-118 (1970). Weierstrass, K. (1) Zur Theorie der eindeutigen analytischen Func- tionen, Abh. Konigl. Akad. Wiss., 1876. Yanagihara, N. (1) Angular cluster sets and oricyclic cluster sets, Proc. Japan Acad. 45, 423-428 (1969). - GENERAL REFERENCES Bagemihl, F. (7) The Lindelof theorem and the real and imaginary parts pi normal functions, Mich. Math. J. 9, 15-20 (1962). meromorphic functign pp sequegces pi Jordan curves, Ann. Acad. Sci. Fenn. Ser. A1 328, 1-14 (1963). (9) Some boundary properties pi normal functions bounded pp nontangential arcs, Arch. Math. 14, 399-406 (1963). (10) Meier points pi.holomorphic functions, Math. Ann. 155, 422-424 (1964). (11) Sets pi asymptotic values pi positive linear measure, Ann. Acad. Sci. Fenn. Ser. AI 373, 1-7 (1965). (12) Chordal limits pi holomorphic functions pi Plessner points, J. Sci. Hiroshima Ser. A-I Math. 30, 109-115 (1966). (13) 9p the sharpness pi_Meier's analogue pi Fatou's theorem, Israel J. Math. 4, 230-232 (1966). (14) J. Analyse Math. 20, 407-413 (1967). 'I (15) Some results and problems concerning chordal principal cluster sets, Nagoya Math. J. 29, 7-18 (1967). (16) Bounded holomorphic functions with given boundary ambigu— ous points, Nieuw Arch. Wisk (3) 16, 165-166 (1968). (17) The chordal and horocyclic principal cluster sets f a certain holomorphic function, Yokohama Math. J. 16, 11- 14 (1968). (18) Meier points and horocyclic Meier points pi continuous functions, Ann. Acad. Sci. Fenn. Ser. A1 461, 1-7 (1970) (19) Tpp principal ppp chordal principal cluster sets pi A certain meromorphic function, Rev. Roumaine Math. Pures Appl. 15, 3-6 (1970). 149 150 Bagemihl, F. and J. E. McMillan (2) Radii pi uniform boundedness ppp indetermination pi holomorphic functions, ppp examples ip conformal mapping pi_Jordan regions, Ann. Acad. Sci. Fenn. Ser. AI 397, 1-14 (1967). Bagemihl, F., G. Piranian and G. Young (1) Intersections pi cluster sets, Bul. Inst. Politehn. Iasi (N. S.) 5 (9), no. 3-4, 29-34 (1959). Barth, K. F. and W. J. Schneider (4) pp A problem pi Collingpood concerning meromorphic functions with pp_asymptotic values, J. London Math. Soc. (2) 1, 553-560 (1969). (5) 9p.p question pi Seidel concerning holomorphic functions bounded pp_p spiral, Canad. J. Math. 21, 1255—1262 (1969). functions, Nagoya Math. J. 40, 213-220 (1970). Brown, L. and P. Gauthier (2) Cluster sets pp p Banach algebra pi non-tangential curves, Ann. Acad. Sci. Fenn. Ser. A1 460, 1-4 (1969). Cargo, G. T. (1) Radial and angular limits pi meromorphic functions, Canad. J. Math. 15, 471-474 (1963). Cartwright, M. L. and E. F. Collingwood (1) The radial limits pi functions meromorphic ip p circular disc, Math. Z. 76, 404-410 (1961). Church, P. T. (2) Global boundary behavior pi meromorphic functions, Acta Math. 105, 49-62 (1961). (3) Boundary images pi_meromorphic functions, Trans. Amer. Math. Soc. 110, 52-78 (1964). Collingwood, E. F. (5) 9p functions meromorphic ip the unit disc and restricted pp p_spira1 pp the boundary, J. Indian Math. Soc. (N. S.) 24, 223-229 (1960). (6) Tsuji functions with Julia points, Contemporary Problems in Theory Anal. Functions (Internet. Conf., Erevan, 177— 179 (1965). (7) A boundary theorem ipp Tsuji functions, Nagoya Math. J. 29, 197-200 (1967). Collingwood, E. F. and G. Piranian (1) Tsuji functions with segpents pi Julia, Math. Z. 84, 246-253 (1964). Doob, J. L. (1) Cluster values pi sequences pi analytic functions, Sankhya Ser. A 25, 137-148 (1963). 151 Erdos, P. and G. Piranian (1) Restricted cluster sets, Math. Nachr. 22, 155-158 (1960). and asypptotic, Studies in mathematical analysis and related topics, 93-103, Stanford Univ. Press, Stanford, Calif., 1962. Faust, C. M. (1) 9p the boundary behavior pi holomorphic functions ip the unit disk, Nagoya Math. J. 20, 95-103 (1962). Fuchs, W. H. J. and W. K. Hayman (1) Ap entire function with assigned deficiencies, Studies in mathematical analysis and related topics, 117-125, Stanford Univ. Press, Stanford, Calif., 1962. Gauthier, P. (3) The maximum modulus pi normal meromorphic functions and applications pp value distribution, Canad. J. Math. 22, 803-814 (1970). (4) Unbounded holomorphic functions bounded pp p spiral, Math. Z. 114, 278-280 (1970). Gavrilov, V. I. (2) 9p the set pi angular limiting values pi normal meromorphic functions, Dokl. Acad. Nauk SSSR 141, 525- 526 (1961). (3) Boundary behavior pi functions meromorphic ip the unit circle, Dokl. Acad. Nauk SSSR 151, 19-22 (1963). circle, Dokl. Acad. Nauk SSSR 148, 16-19 (1963). (5) Limits pi_meromorphic and generalized meromorphic functions along continuous curves and sequences pi points ip the unit circle, Vestnik Moskov. Univ. Ser. I Mat. Meh., no. 2, 30-36 (1964). (6) Ambiguous points pi_meromorphic functions, Vestnik Moskov. Univ. Ser. I Mat. Meh., no. 4, 29-34 (1965). (7) Boundary behavior pi functions meromorphic ip ppp unit circle, Vestnik Moskov. Univ. Ser. I Mat. Meh., no. 5, 3-10 (1965). (8) Certain boundapy theorems pi uniqueness ipp meromorphic functions, Uspehi Mat. Nauk 20, no. 6, 59—63 (1965). (9) A remark pp.ppp asypptotic behavior pi holomorphic functions, Sibirsk. Mat. Z. 7, 212-216 (1966). Moskov. Univ. Ser. 1 Mat. Meh. 21, no. 6, 25-27 (1966). 152 (ll) Qp p uniqueness theorem, Nagoya Math. J. 35, 151-157 (1969). Goffman, C. and W. Sledd (1) Essential cluster sets, J. London Math. Soc. (2) 1, 295-302 (1969). Hall, R. (1) Qp the asymptotic behavior pi functions holomorphic ip the unit disc, Math. Z. 107, 357—362 (1968). Hayman, W. K. (3) Regular Tsuji functions with infinitely many Julia points, Nagoya Math. J. 29, 185-196 (1967). (4) The boundapy behavior pi Tsuji functions, Mich. Math. J. 15, 1-25 (1968). Heins, M. (l) Qp the boundary behavior pi p conformal map pi the open unit disk into p Riemann surface, J. Math. Mech. 9, 573-581 (1960). Kaczynski, T. J. (3) Boundary functions for bounded harmonic functions, Trans. Amer. Math. Soc. 137, 203-209 (1969). Kate, M. (1) pp the boundary behavior pi_meromorphic functions ip the unit circle, Rep. Lib. Arts Sci. Fac. Shizuoka Univ. Sect. Natur. Sci. 3, 249-254 (1965). Kegejan, E. M. (1) Cluster sets pi analytic functions defined ip p disc, Acad. Nauk Armjan. SSSR Dokl. 43, no. 1, 6-11 (1966). Kuramochi, Z. (1) Cluster sets pi analytic functions ip open Riemann surfaces with regular metrics, Osaka Math. J. 11, 83-90 (1959). Lappan, P. and D. C. Rung (1) Normal functions and non-tangential boundary arcs, Can. J. Math. 256—264 (1966). Lohwater, A. J. (1) The cluster sets pi_meromorphic functions, Treizieme congres des mathematicians scandinaves, tenu a Helsinki, 171-177 (1957). Mercators Tryckeri, Helsinki, 1958. (2) 1p; exceptional values pi_meromorphic functions, Colloq. Math. 7, 89-93 (1959). (3) 9p ppp theorems pi Gross ppp Iversen, J. Analyse Math. 7, 209-221 (1959/60). MacLane, G. R. (3) Meromorphic functions with small characteristic ppp pp asypptotic values, Michigan Math. J. 8, 177-185 (1961). (4) Holomorphic functions, pi arbitrarily slow growth, without radial limits, Michigan Math. J. 9, 21-24 (1962). 153 Mathews, H. T. (3) Left and right boundary cluster sets ip n-space, Duke Math. J. 33, 667-672 (1966). (4) The n-arc property for functions meromorphic ip the disk, Math. Z. 93, 164-170 (1966). Mathews, J. H. (3) A bounded normal light interior function that possesses pp_point asymptotic values, Israel J. Math. 7, 381-383 (1969). Nagoya Math. J. 39, 149-155 (1970). (5) Coefficients pi uniformly normal-bloch functions, Yokohama Math. J. 21, 27-31 (1973). —.—.-————__.——._.—_ 39, 274-277 (1963). (2) Some notes pp the cluster sets pi meromorphic functions, Proc. Japan Acad. 42, 1027-1032 (1966). McMillan, J. E. (4) Asymptotic values pi functions holomorphic ip the unit disc, Michigan Math. J. 12, 141-154 (1965). (5) 9p local apymptotic properties, ppp asymptotic value sets, App ambiguous properties pi functions meromorphic ip ppp open unit disc, Ann. Acad. Sci. Fenn. Ser. AI no. 384, 1-12 (1965). (6) Curvilinear oscillations pi holomorphic functions, Duke Math. J. 33, 495-498 (1966). (7) 9p cluster sets pi meromorphic functions, Proc. Nat. Acad. Sci. USA 56, 787-788 (1966). (8) Qp_metric properties pi_sets pi_angu1ar limits pi meromorphic functions, Nagoya Math. J. 26, 121-126 (1966). (9) pp the asymptotic behavior pi functions harmonic i __ p disc, Nagoya Math. J. 28, 187-191 (1966). _____—.—_—___——__ 91, 186-197 (1966). (11) A boundary property pi holomorphic functions, Math. Ann. 173, 275-280 (1967). (12) Boundary properties pi analytic functions, J. Math. Mech. 17, 407-419 (1967). (13) Open mappings and cluster sets, Rev. Roumaine Math. Pures Appl. 12, 1079-1086 (1967). 154 (14) Cluster sets pi meromorphic functions, Proc. Amer. Math. Soc. 23, 148-150 (1969). (15) pp the asypptotic values pi p holomorphic function with nonvanishing derivative, Duke Math. J. 36, 567-570 (1969). Njastad, O. (l) Infinite-valued asymptotic ppints and Koebe arcs, Math. Scand. 19, 172-182 (1966). (2) Infinite-valued asypptotic points for functions holomorphic ip the unit disc, Norske Vid. Selsk. Skr. (Trondheim), no. 8, 1-14 (1966). Noshiro, K. (3) Cluster sets pi pseudo-analytic functions, Japan J. Math. 29, 83-91 (1959). (4) Some theorems pp cluster sets, Ann. Acad. Sci. Fenn. Ser. AI no. 389, 1-8 (1966). (5) Some remarks pp cluster sets, J. Analyse Math. 19, 283-294 (1967). (6) Some theorems pp cluster sets, Hung-ching Chow Sixty-fifth Anniversary Volume, Math. Res. Center Nat. Taiwan Univ., Taipei, 1-6 (1967). Piranian, G. (1) pp A problem pi Lohwater, Proc. Amer. Math. Soc. 10, 415-416 (1959). Plesner, A. I. (1) Behavior pi analytic functions pp the boundapy pi their region pi definition, Uspehi Mat. Nauk 22, no. 1 (133), 125-136 (1967). pp arcs pi positive hyperbolic diameter, J. Math. Kyoto U. 8, 417-464 (1968). Ryan, F. (1) A characterization pi the set pi asypptotic values pi p function holomorphic ip the unit disc, Duke Math. J. 33, 485-493 (1966). (2) The set pi asypptotic values pi p bounded holomorphic function, Duke Math. J. 33, 477-484 (1966). Stebbins, J. (2) pp the Riemann surface generated py_p function meromorphic ip the unit disk with pp p spiral asypptotic value, Math. Z. 96, 179-182 (1967). (3) Spiral asypptotic values pi functions meromorphic ip the unit disk, Nagoya Math. J. 30, 247-262 (1967). Storvick, D. A. (1) Cluster sets pi pseudo-meromorphic functions, Nagoya Math. J. 18, 43-51 (1961). 155 (2) Radial limits pi quasiconformal functions, Nagoya Math. J. 23, 199-206 (1963). Suzuki, J. (1) 9p asypptotic values pi slowly growing algebroid functions, Nagoya Math. J. 41, 135-148 (1971). Tanaka, C. (1) Note on the cluster sets of the meromorphic functions, ———.—___.._—_.._._—__ Proc. Japan Acad. 35, 167-168 (1959). Tse, K. F. (2) 9p the sums and products pi normal functions, Comment. Math. Univ. St. Paul. 17, 63-72 (1969). Vessey, T. A. (2) Some properties pi oricyclic cluster sets, J. Analyse Math. 21, 373-380 (1968). Weiss, M. L. (l) Cluster sets bounded analytic functions from p Banach algebraic viewpoint, Ann. Acad. Sci. Fenn. Ser. AI no. 367, 1-14 (1965). Woolf, W. B. (1) The boundary behavior pi meromorphic functions, Ann. Acad. Sci. Fenn. Ser. AI no. 305, 1-11 (1961). Yamashita, S. (l) Cluster sets pi algebroid functions, Tohoku Math. J. (2) 22, 273-289 (1970). (2) Some theorems pp cluster sets pi set-mappings, Proc. Japan Acad. 46, 30-32 (1970). Yoshida, H. (1) A remark pp Plessner points, J. Fac. Engrg. Chiba Univ. 20, 153-154 (1969). Young, G. S. (1) iypes pi ambiguous behavior pi analytic functions, Michigan Math. J. 10, 147-149 (1963). RECENT REFERENCES APPLICABLE TO FUTURE RESEARCH Dragosh, S. (3) Koebe sequences of arcs and normal functions, Trans. Amer. Math. Soc. 190, 207-222 (1974). Kurbanov, K. O. (1) 9p uniformly normal meromorphic functions, Moscow University Mathematics Bulletin 28, no. 5-6, 67-69 (translated from Russian from Vestnik Moskovskogo Universiteta Matematika 28, no. 6, 18-20 (1973) ). functions, Proc. Amer. Math. Soc. 44, 403-408 (1974). (15) A criterion for p_meromorphic function t __ pp normal, Comm. Math. Helv. (to appear). Lohwater, A. J. and Ch. Pommerenke. (1) pp normal meromorphic functions, Ann. Acad. Sci. Fennicae AI 550, 1-12 (1973). Stebbins, J. (4) Boundary functions nd sets of asypptotic values, Journal of Approximation Theory 6, 421-430 (1972). 156 l , . ...s .. . if...