ABSTRACT NORMALCY OF SUMS AND PRODUCTS OF NORMAL FUNCTIONS AND REAL AND COMPLEX HARMONIC NORMAL FUNCTIONS By David Winfield Bash Jr. Let f and g be normal functions from the unit disk into the Riemann Sphere. A normal function is a uniformly continuous function when using the hyperbolic metric in the disk and the chordal metric in the Sphere. Necessary and sufficient conditions are established for f+g to be normal and for fg to be normal. From these conditions some simpler sufficient conditions for f+g to be normal and fg to be normal are established. Also inequalities and quotients involving normal functions are investigated. Employing the product results several theorems concerning normal functions and Bers' pseudo- analytic functions of the first kind are presented. Real valued normal functions are investigated and used to obtain information about normal functions in general. The logarithm of a normal function, if continuous and single valued, is normal and conditions guaranteeing that the exponential of a normal function is normal are developed. 'Using the exponential function a necessary and sufficient condition for a real harmonic function to be not normal is proved and applied to show that if u and v are real har- monic, normal and u-K. < w < v+K , then w harmonic is normal. The exponential function is also used to connect sums of real normal functions and products of certain normal functions. David Winfield Bash Jr. A subclass of the class of normal functions is defined and characterized. The functions of this subclass, called very normal, are closed under additions and projections to the real and imaginary axes and they are compared with uniformly normal functions. A type of sequence, called (A) sequence, is defined for an arbitrary function f and it is shown that the complex valued harmonic function f is not normal if and only if f has a special type of (A) sequence. Also (A) sequences are used to obtain several other properties of functions which are, in some sense, not normal. NORMALCY OF SUMS AND PRODUCTS OF NORMAL FUNCTIONS AND REAL AND COMPLEX HARMONIC NORMAL FUNCTIONS By David Winfield Bash Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1969 957/1/ //-;4~L7 To Nancy and To Mom and Dad ii ACKNOWLEDGEMENT I wish to thank Professor Lappan for his thoughtful sugges- tions and insights on the behavior of normal functions. iii 1 WW‘IT’mm '1 3 . vh' l"_" ' a‘fi'}; ‘ 'fizJIIE' ,b kl - .- wrtd g . . 1 II. III. IV. INTRODUCTION TABLE OF CONTENTS OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO SUMS AND PRODUCTS OF NORMAL FUNCTIONS .............. REAL NORMAL FUNCTIONS .............................. VERY NORMAL FUNCTIONS .............................. (A) SEQUENCES FOR NOT NORMAL FUNCTIONS ............. BIBIJOGRAPHY iv 16 26 31 40 I. INTRODUCTION Lehto and Virtanen [15, p. 47] defined the function f(z) to be normal as follows: If f(z) is meromorphic in a simply connected domain G, then f(z) is normal if and only if the family {f(S(z))} is normal in the sense of Montel where z' = 8(2) denotes an arbitrary one-one conformal mapping of G onto itself. Lappan in [13] essen- tially defined a complex valued function f(z) to be normal in a simply connected hyperbolic region G if and only if f(z) is uniformly continuous using the hyperbolic metric in G and taking as the range of f(z) the Riemann Sphere W with chordal metric. Let D be the unit disk and C the unit circle. DEFINITION 1.A. The hyperbolic distance, p(zl,zz), between the points 21 and 22 in D is given by (1/2)log((1+u)/(l-u)) = tanh-lu, where u . ‘21-22‘/|1-5122|. (See [9, Chapter 15].) Since the hyperbolic distance between points in a simply connected hyperbolic domain G is defined to be the hyperbolic‘dis- tance between their images in D under any conformal map from G onto D, we shall usually have D as the domain of our functions. Now we state more precisely Lappan's definition of a normal function. DEFINITION 1.3. [13, p. 156]. A complex valued function f(z) in D is a normal function if and only if, for each pair of sequences {2“} and {2;} of points in D such that p(zn,z;) a O, the convergence of {f(zn)} to a value a in W implies the convergence of {f(z$)} to a. In the case of meromorphic functions the above definition is equivalent to Lehto and Virtanen's definition of normality (also see theorem 3.A). We now state a few definitions and theorems we will use frequently in this thesis. DEFINITION 1.C. (Lappan [12, p. 43].) Two sequences {2n} and {2;} of points in D such that p(zn,z;) a O as n q»m are called close sequences. DEFINITION 1.D. (Lappan [12, p. 44].) The meromorphic function f(z) in D is said to have property (D) on a sequence {2n} of points in D if [f(zn)} converges and, for each complex number 6, there exists a sequence {25] close to {2“} such that f(zé) a 6. THEOREM l.E. (Lappan [12, p. 45].) The meromorphic function f(z) .i_r_1_ D is not normal _i_i: and oily if there exists a sequence 9_f_ points in D on whigh_f(z) has property (D). DEFINITION 1.F. (Lappan [12, p. 46].) A holomorphic function f(z) in D is said to be uniformly normal if, for each M, there exists a finite number K such that, for each 20 in D, p(z,zo) < M.imp1ies ‘f(z)-f(zo)‘ < K. (THEOREM 1.G. (Lappan [12, p. 46].) A uniformly normal holomorphic function l§.2 normal function. [DEFINITION l.H. By the cluster values of a function f(z) on a sequence {2n} of points in D we mean those complex numbers a for which there exists a subsequence {zn'] of {Zn} such that f(z )aaask-oon. “k A function in a Hardy p-class, p > O, or a function of bounded characteristic can be written as a sum or product of two normal functions, but sums and products of normal functions need not be normal (see Lappan [11]). It would be desirable to know when the sum or product of two or more normal functions is normal. The basic results in this direction are: (1) if f(z) is normal and meromorphic in D, g(z) a bounded holomorphic function in D, then f(z)+g(z) is normal in D (Lehto and Virtanen [15, p. 53]); and (2) if f(z) is normal and meromorphic in D, g(z) a holomorphic function in D such that O < M1 < ‘g(z)“< M < m, then f(z)g(z) is 2 normal in D (Lappan [11, p. 188]). The sum of two uniformly normal functions is normal (Lappan [12, p. 46]) but the product of uniformly normal functions may not be normal. For example, let f(z) = Log(l-z) and let Bf(z) be a Blaschke product such that f(z)Bf(z) is not normal (see Lappan [12, p. 48] and [11, p. 190]). In chapter two some necessary and sufficient conditions for sums and products of normal functions to be normal and sufficient conditions for functions bounded by normal functions to be normal will be presented. In an effort to obtain a better understanding of holomorphic normal functions, chapter three investigates normal real valued harmonic functions (we drop the property of being harmonic when convenient) and their relationships to normal complex valued functions and some simple expressions involving normal functions. Chapter four develops a subclass of the class of normal functions which are closed under addition, and chapter five investigates normalcy and non-normalcy of real and complex harmonic functions through behavior on close sequences. II. SUMS AND PRODUCTS OF NORMAL FUNCTIONS We start with some theorems on sums and products of normal functions whose proofs employ Lappan's definition of normalcy through convergence on close sequences. THEOREM 2.1. Let f(z) and g(z) 39 normal functions _i_r_1_ D. Then f(z)+g(z) l3 normal lE.D ii and only if for each sequence {2n} ip.D such that f(zn) d'w, g(zn) ~»m, and {f(zn)+g(zn)} converges Egmg complex value a (possibly a), the sum {f(zg)+g(zg)} converges to a for each sequence {2;} glpgg pp {2n}. ‘gpppf. The necessity is obvious from the definition of normal functions. For sufficiency, let [Zn] be a sequence where f(zn)+g(zn) » a. By considering appropriate subsequences, if necessary, we may assume that f(zn) a B and g(zn) « y, 9 and y complex numbers. Let {2;} be close to {Zn}. If B ==c = y, then f(z;)+g(z;) a a by the condition of the theorem. Otherwise B+y is a well-defined complex number (possibly a) and by the normalcy of f(z) and g(z) we get that g(zé)+g(z;) a 3+y. Hence f(z)+g(z) is normal. Using theorem 2.1 we can prove several sufficient condi- tions for f(z)+g(z) to be normal in D where f(z) and g(z) are normal in D. COROLLARY 2.2. If f(z) and g(z) are normal 1p.D and there i .3 number M5 0 < M«< a, such that the sets {z:|f(z)‘ >'M] and {z:‘g(z)‘ >tM} are disjoint; then f(z)+g(z) i§_normal ip_D. 4 ‘gpppf. The hypotheses of the condition of theorem 2.1 are satisfied vacuously. Let f(z) and g(z) be holomorphic functions in D which omit the value zero, and let Qn(f) = {2: n < [f(z)] < 2n}, Rn(f,g) = Qn(f) n {2: l—l/ng < [f(z)/g(z)] < 1+l/nk] and Tn(f,g,p) = {2: ‘arg f(z)-arg g(z)] s n-u(n)] where g(n) is a non- increasing function on the positive integers such that 0 < u(n) < n. For each 2 the arguments of f(z) and g(z) are chosen so as to have the difference, in absolute values, less than or equal to n. Hence the "arguments" in this chapter are not continuous functions of z. COROLLARY 2.3. L3; f(z) £29 g(z) pg_normal holomorphic functions ip_D, gg£h_p£ which omits the value zero, and u(n) ppgh EEEE (n sin u(n))/(2 cos(u(n)/2)) increases £2 infinity mono- tonically g§,n increases pp infinity. lf_£hg£g_i§ 3 positive integer N ppghflghgp n 2 N implies Rn(f,g) is contained 22. Tn(f,g,u), £222 f(z)+g(z) is‘a normal holomorphic function ip_D. ‘gpppf. It suffices to show that if [2m] is a sequence of points in D and f(zm) a a, then f(zm)+g(zm) ~»m. Let c > 0 be given and let N satisfy the hypotheses, and let m > P 2 N and both %-1)) > 1/3. (P sin p.(P))/(2 cos(p,(P)/2)) > Us and P(l/(P We may assume without loss of generality that zm is in Qm(f). Either zm is in Rm’ or \f(zm)/g(zm)‘ s 1-l/mg, or [f(zm)/g(zm)‘ 2 1+1/mg. If zm is in Rm’ then zm is in Tm(f,g,u). Since [g(zm)‘ > (1/2)m for m > 4, by computing the length of the diagonal of the parallelogram of sides m and (l/2)m, with interior angles g(m) and n-p(m), and noting that this diagonal is shorter than |f(zm)+g(zm)‘, we see that |i(zm)+g(zm)| > (m sin n(n))/(2 cos(n(m)/2)) > (P sin n(p))/(2 cos(u(P))/2) > I/e. If \f(zm)/g(zm)‘ s l-l/mk = agil, then \f\ s </m$’>\gl <|g|and \g\ > (m5 So \f(zm)+g(zm)| 2 |g(2m)| - [f(zmfl 2 ‘f(zm)‘(m%/(m35 % /(m%-l))‘f(zm)‘. -1)-l) > m(1/(m%-1)) > P(1/(P -1)) > l/e- If |f(zm)/g(zm)\ 2 1+1/m%, then ‘f(zm)+g(zm)‘ > l/e similarly. We note that the % in the definition of Rn(f,g,u) could be replaced by any number between zero and one to obtain a similar result. In the proof we show that if f(zm) ~ a, then f(zm)+g(zm) a a also. In doing this, we use the arguments of f(z) and g(z) only when ‘f(z)l and \g(z)‘ are large and hence we don't need to require that f(z) and g(z) be non-zero. If f(z) and g(z) are meromorphic, we exclude those {Zn} in D which are poles of f(z) or g(z) and consider neighborhoods [Um] in D of {Zn} such that Pi< [f(z)] < a, g(z) # a for all z in Un - [Zn], where P is as in above proof. As before with given 3 > O, ‘f(z)+g(z)\ > 1/3 for every 2 in Un - [Zn]. Since f(z) and g(z) are continuous at zn, \f(zn)+g(zn)| 2 1/3. There exist normal holomorphic functions f(z) and g(z) whose sum is normal but the functions fail to satisfy the hypotheses of corollary 2.3. Hence the hypotheses of corollary 2.3 are not necessary. For example, let f(z) = (z+1)/(z-l), and g(z) = (-l)(z+l)/(z-1) + z in D. Then f(z)+g(z) = 2 which is normal; but, as evident, there is no positive integer N such that n 2 N implies Rn(f,g) is contained in Tn(f,g,u) for any u(n) satisfying the hypotheses. COROLLARY 2.4. Let f(z) and g(z) pg normal holomorphic functions ip_D. If there exist 6, M, t, R where O < 5 < n, l < t < 2, 0 < Mt< m, 0 S R.< 1 such that the set {zz Mi< [f(z)] < a} n {2: l/t < [f(z)/g(z)] < t} n {z: R < ‘2‘ < 1} i§_contained i2 {2: ‘arg f(z)-arg g(z)‘ S n - 5]; then f(z)+g(z) “is g normal holomorphic function 33 D. COROLLARY 2.5. Let f(z) and g(z) pg normal holomorphic functions ip_D. If there exist 5, M, R where O < 6 < n, O < M‘< a, O S R < 1 such that the set {2: M‘< ‘f(z)| < m] n {2: M'< [g(z)‘ < m} n [2: R < ‘2‘ < l] is 222‘ ptained in [2: ‘arg f(z)-arg g(z)| S n - 6]; then f(z)+g(z) is‘a normal holomorphic function ip_D. As before, the arguments of f(z) and g(z) in corollaries 2.4 and 2.5 are chosen so as to have the difference, in absolute values, less than or equal to n. Then the corollaries 2.4 and 2.5 follow easily from corollary 2.3. THEOREM 2.6. LEE f(z), g(z), a(z), 33d b(z) pg normal holomorphic functions ip,D High a(z)f(z) £29 b(z)g(z) normal, a(z) 229 b(z) uniformly bounded ip.D py.A Egg B respectively. LEE Dn = {z: n < \f(z)‘ < a, n < [g(z)] < a, ‘2‘ > 1-1/n} 33g En = [2: ‘arg a(z)-arg b(z)] S a(n)}, yhgpg a(n) ip'g real valued function pp the ppsitive integers such that u(n)-B(n) lg pppjip- creasing, O S u(n)-B(n) S n, y1£h_u(n) gpd'Tn(f,g,u) E§.ifl corollary 2.3. ._f n sinQ1(n)-B(n)) montonically increases pp infinity with n and there exists 3 positive integer N such that DN O (D ‘ (EN U TN(f,g.u))) 1§_empty, then a(z)f(z) + b(z)8(2) ii 3 normal holomorphic function 33 D. Proof. Let N* = max(AN,BN,N) and let 2 be in {z: N* < ‘a(z)f(z)‘ < m, N* < ‘b(z)g(z)‘ < a, ‘z‘ > 1-1/N*}. Then z is in DN so 2 is not in D - (EN U TN(f,g,u)). Hence ‘arg a(z)f(z)-arg b(z)g(z)‘ S n - (u(N)-B(N)) and we may complete the proof by applying corollary 2.5 to a(z)f(z) and b(z)g(z) with M = N*, R = l-l/N*, and 6 = u(N)-B(N). g3._l I min We can also get a necessary and sufficient condition for E - a product of normal functions to be normal by considering again a “p n r}: convergence on close sequences. [E g, THEOREM 2.7. LEE f(z) 32d g(z) pg normal functions ip_D. ngp f(z)g(z) i§_normal 13 D i£_gpg.pply if for each sequence {2n} _i__r;D $131523: f(zn) - 0. g(zn) -° 0° (2; f(zn) -* m, g(zn) -* 0) 31% {f(zn)g(zn)] converges pp p_complex value a (possibly a), pp; product {f(z;)g(z;)] converges £9_a for each sequence {2;} plpgg £2 {2,1» gpppf. The necessity is immediate. For sufficiency, let {2n} be a sequence where f(zn)g(zn) a a. By considering appropriate subsequences, if necessary, we may assume that f(zn) a B and g(zn) a y, B and y complex numbers. Let {zé} be close to {Zn}. If B = O, y = a, or if a = m, y = O, we have f(zg)g(z;) a a from the condition of the theorem. Otherwise By is a well defined complex number (possibly a) and by the normalcy of f(z) and g(z), f(z;)g(zé) a By also. Hence f(z)g(z) is normal in D. The following corollary follows easily from theorem 2.7. COROLLARY 2.8. Let f(z) and g(z) pg normal functions 12 D. 1;: there exists finite positive constants Kf, Kg’ Mf, Mg such that the sets D1 = [2: ‘f(z)‘ > Kf; ‘g(z)‘ > Kg} 33d. D2 = {z: ‘f(z)‘ < Mf; ‘g(z)‘ < Mg} cover D (i.e. D = D1 U D2), then f(z)g(z) ig‘g normal function ip_D. Also the hypotheses of the following theorem of Zinno [18, pp. 160-161] vacuously satisfy the hypotheses of theorem 2.7. THEOREM 2.A. Let f(z) and g(z) 22 two normal meromorphic functions ip D. Let av and a; pg zeros pf f(z) and g(z) respectively and let bv and b; 23 poles pf f(z) and g(z) respectively. Su ose that a ' o I (1) inf p(av,bu) > O and inf p(av,bu) > O v= . v=1,2,. u: 3 3 ,. u=1,2,. l 2 1,2 (inf over pp empty set is infinite) and (2) for any positive number p there exists a positive Mm insignia m ‘£(z)| < mp g3; z ip_D - J:1U(bv,p) a) lg| < mp £25 2 2D - U u v=1 v Q ‘f(z)‘ > l/m for z in D - U U(a ,p) p ___ .—_ v=1 v and ‘g(z)‘ > l/m for z in D - u U(af,p) p — —- v=1 V (U(Z,6) = {Ci 9(Z:C) < 5}) Then the product f(z)g(z) i§.g normal meromorphic function 13 D. By excluding the possibility of sequences satisfying pro- perty (D), we can get another condition to guarantee the sum of two normal meromorphic functions to be normal and get some ideas on what is sufficient for functions bounded by normal functions to be normal. Again in the following theorem the arguments of the 10 functions f(z) and g(z) are chosen so as to have the difference in absolute values less than or equal to n. THEOREM 2.9. LEE f(z) 33d g(z) 23 two normal meromorphic functions ip_D. LEE 6 > Opp; arbitrary and for each sequence {2n} 'fgpflyhigh f(zn) a a, f(zn) # m, g(zn) # a, let there exist a(n), lg non-increasing function on the positive integers, such that 0 S a S n and ‘f(zn)‘sin a(n) 2 6 for all n greater than some positive integer N. .lf l-l/‘f(zn)‘ < ‘f(zn)/g(zn)‘ < 1-+l/‘f(zn)‘ implies ‘arg f(zn)-arg g(zn)‘ S n-a(n) fp£_n > N, phgp f(z)+g(z) ighg normal meromorphic function 13 D. PEpof. It is clear that if [2“] is a sequence where {f(zn)} has no unbounded subsequences and {f(zn)+g(zn)] converges, then {2n} is not a sequence for which f(z)+g(z) has property (D). Hence let {2n} be a sequence for which {f(zn)} diverges to a and {f(zn)+g(zn)] converges to a finite complex value or diverges to a. Let {2;} be any sequence close to {an} so that f(za) a m also. Fix n > N and consider the following cases. Case I. f(zg) # o. (a.) If g(zr'l) = s, then ‘g(zr'l)+f(zr'l)‘ = s. (b.) If g(zé) #=¢ and 1-1/‘f(zr'l)‘ < ‘f(zr'l)/g(zr'1)‘ < l+l/‘f(zr'1)‘, then ‘f(zt'1)+g(zr'l)‘ 2 ‘f(zr'l)‘sin a(n) 2 e > o for n > N, the 5 and N in the hypotheses of the theorem. (c.) If g(zr'l) #s and ‘f(zr'l)/g(zr'l)‘ 2 l+1/‘f(zr'l)‘, then lg(zg)\ s (If(z;>|/(1+lf(z;)|))|f(z;)|. Hence |f(z;)+s(z;)\ 2 \f(z;>| - ‘g(z;>| 2 ll |fl(1-‘f\/<1+\f|>> = 1/(1+1/‘f(z;)‘) 2 1/2 > O for n sufficiently large. (d.) If g(zé) #:e and ‘f(z$)/g(z$)‘ s l-1/‘f(z;), then (‘f(z;)‘/(‘f(zé)‘-l))‘f(zé)‘ s ‘g(z;)‘. Hence \f(z;>+gl 2 lgl - |f(z;>l 2 ‘f(z;)‘(‘f(z;)‘/(‘f(z;)‘-l)) = 1/(1-1/‘f(z;)‘) 2 l > O for n sufficiently large. Case II. f(zé) = a. (a.) If g(zé) # s, then ‘f(zé)+g(z;)‘ = m. (b.) If g(zé) = m, then there exists a neighborhood U containing 2; such that f(z) # m and g(z) # m for o . a: all z in U - {z;]. Then there ex1sts {zn,m}m=1 con- tained in U, z w 2' as m.S m, where ‘f(z )‘ > m n,m n n,m and we may proceed as in case 1. Thus there is no sequence on which f(z)+g(z) has property (D). We know that a meromorphic function bounded above (or below) by a normal meromorphic function is the product and sum of normal meromorphic functions. We can say more than this if the function is appropriately bounded both above and below. THEOREM 2.10. LEE f(z), g(z), 32d h(z) pg meromorphic functions i3 D such that g(z) and h(z) are normal; and let K1 and K2.E§ Constants such that ‘g(z)‘-K1 S ‘f(z)‘ S ‘h(z)‘+l(2 for each z 12 D. ‘lf lim inf ‘g(zn)‘ > K1 for each sequence {2“} ip'D for n—m which h(zn) S.¢, then f(z) is normal 1E.D' Proof. Assume f(z) is not normal. Then there exists a sequence {an} such that f(z) has property (D) on {2“}. Then {Zn} has a subsequence {Zn ] for which there exist sequences [2; } and k 12 {2" ], both close to {z }, satisfying f(z' ) a O, f(z" ) a Q. But “k ”k “k “k then lim inf ‘g(z; )‘ S K1 and h(z" ) ~»m. Since g(z) and h(z) are ndm k nk normal, this means that lim inf ‘g(z )‘ S K and h(z ) a m, in n l n nda k k violation of the hypotheses. Thus there is no sequence {Zn} on which f(z) has property (D), and the theorem is proved. REMARK 2.11. Since there are non-normal functions of bounded characteristic the condition above on sequences is needed. For example, Bagemihl and Seidel [4, p. 7] construct a holomorphic function f(z) of bounded characteristic such that f(zn) = O for 2n = l-l/n2 and f(zé) a m where z; = (1/2)(zn+zn+1) and an ele- mentary calculation shows that {Zn} and {2;} are close. Hence f(z) is not normal. Their function is B(z)/exp((-l-z)/(1-z)) where B(z) = a(zn-zVu-znz). But ‘B(z)‘ s ‘f(z)‘ s ‘1/exp((-1-z)/(1-z))‘, and B(z? and l/exp((-1-z)/(1-z)) are normal holomorphic functions. Since B(zg) a O and 1/exp((-l-z;)/(l-z;)) a a, there exists a sequence, namely {2;}, not fullfilling the hypotheses of theorem 2.10. COROLLARY 2.12. LEE f(z) gpg_h(z) pg meromorphic functions 12D £151‘h(z)‘-K1 s ‘f(z)‘ s ‘h(z)‘+l(2. Inge f(z) i_s_ normal 1: 322 pply if h(z) 1g normal. THEOREM 2.13. Lg f(z) = h1(z)/h2(z) E the quotient 9i Ego bounded holomorphic functions ip_D yi£h_pp common zeros. If f(z) is not normal, then there igua sequence [Zn] of points ip_D seer; shes h1 ~ 0 222 h2 -. o. .EEQQE: We prove the contrapositive. Let M1 2 ‘h1(z)‘ and M2 2 ‘h2(z)‘. Then ‘h1(z)/M2‘ s ‘f(z)‘ s ‘Ml/h2(z)‘ and h1(z)/M2 and Ml/h2(z) are normal. Since there is no sequence {Zn} where h1(zn)/M2 » 0 and Ml/h2(z) ~»m, f(z) is normal by theorem 2.10. l3 COROLLARY 2.14. ‘Lep h1(z) = B1(z)exp(g1(z)), h2(z) = B2(z)exp(g2(z)) pe_holomorphic functions with B1(z) 321 B2(z) Blaschke products; and M, O < M«< m, gppp_£pgp -M S Re(g1(z)-g2(z)) s M. £23 Li(5) = {2: ‘Bi(z)‘ < 5}, i=1,2. _£_there exists 5' §p£p_pp3p p(L1(6'),L2(6')) = inf {p(zl,22): 21 in L1(6’), 22 in L2(6')}2n>0, £222 f(z) = h1(z)/h2(z) lg EQEEEI° The following corollary gives a restatement of a result of Cima [7, p. 769]. COROLLARY 2.15. le£_F(z) = B1(z,an)/Bz(z,bn), B1(z,an) 329. B2(z,bn) are Blaschke products with zeros {an} and {bu} respectively. 1;; the cluster ppints pf‘ppe'ge£p§.pf|31(z,an) 229 Bz(z,bn) EEE.§£§‘ joint, then F(z) lg normal. Theorem 2.7 may be applied to obtain some properties of normal pseudoanalytic functions in D. The following material may be found in detail in [5] and [6]. DEFINITION 2.3. [5, p. 18]. Let w(z) be a function from D into the complex w-plane which possesses continuous partial deriv- atives. If there exists a constant R such that ‘w2(z)‘ S K‘w(z)‘, we say w(z) is approximately analytic. Let Do be a domain containing D. DEFINITION 2.c. [6, p. 215]. Two continuous functions F(z), G(z) defined in D0 are said to form a generating pair if Im{F(z)G(z)] >'O for z in Do. Every function w(z) defined in D admits the unique re- Presentation w(z) = ¢(z)F(z)+Y(z)G(z) with real functions ¢(z), Y(z), [6, p. 216]. 14 DEFINITION 2.D. [6, p. 217]. The function w(z) = ¢(z)F(z)+Y(z)G(z) is said to possess at the point 20 in D . flgF G)W(z) the (F,G)-derivative, denoted w(zo) or 52 2:2 if the limit 0 , w(z)‘¢(ZO)F(Z)'Y(ZO)G(Z) w(zo) = lim _ exists and is finite. z-o o z 20 DEFINITION 2.E. [6, p. 219]. A function w(z) possessing an (F,G)-derivative at all points of the domain D is called regular (F,G)-pseudoanalytic p£_the first kind lfl.D or simply pseudoanalytic if there is no danger of confusion. THEOREM 2.F. [5, p. 18]. Evepy pseudoanalytic function lg also epproximately analytic. DEFINITION 2.6. [5, p. 24]. We call two functions w(z) and f(z) defined in D similar if there exists a function S(z) which is continuous and different from zero on the closure of D and such that f(z) = S(z)w(z) in D. THEOREM 2.H. [5, p. 24-25]. SIMILARITY PRINCIPLE Every approximately analytic function w(z) lp_D possesses §_similar function f(z) gplpp_l§ analytic lp D. THEOREM 2.16. ll f(z) and w(z) are similar functions lp 2222;, Let S(z) be such that f(z) = S(z)w(z) where S(z) is continuous non-zero on the closure of D. There exists an M such that l/M < ‘S(z)‘ < M for all z in the closure of D. Hence theorem 2.7 implies f(z) is normal if and only if w(z) is normal since S(z) and l/S(z) are normal in D. COROLLARY 2.17. ll_w(z) lg approximately analytic and bounded (pp bounded from zero) lp_D, then w(z) lg normal $2.D“ 15 ‘gpppg. Let f(z) be the analytic function in D similar to w(z). Then f(z) = S(z)w(z) is bounded (or bounded from zero) in D. Thus f(z) is normal and so is w(z) normal. COROLLARY 2.18. If w(z) lpflp normal approximatply analytic function lp'D and has pp asymptotic value a pp_g pp C, then w(z) has p Fatou value 0 pp g‘pp C. Ppppg. Let F be the Jordan arc in D U [Q] on which w(z) tends to the limit a. Let f(z) be the analytic function similar to w(z). Since S(z) in f(z) = S(z)w(z) is uniformly continuous on D U C, S(z) tends to a limit, say 8: (finite and non-zero) on F also. Hence f(z) tends to a limit, Ba, on F and so has a Fatou value Ba at g on C. S(z) also has 5 as a Fatou value at Q. There- fore, since w(z) = f(z)/S(z), w(z) has a Fatou value alat Q. We could show, with proofs similar to those above, that normal approximately analytic functions have many of the pro: perties related to Fatou values of normal analytic functions and that normal pseudoanalytic functions have some of the same identity and uniqueness properties as normal analytic functions. III. REAL NORMAL FUNCTIONS It is easy to see that some simple expressions of normal functions are still normal functions. For example: LEMMA 3.1. Let f(z) pp‘p normal function lp_D into the g-plane. ll ¢(§) lp.p continuous function pp the closure p£_f(D) lppplppp’w-plane, then the function w(z) = ¢(f(z)) mapping D lppp ppp w-plppp.lp normal, 2522£° Let {2“} and {zg} be any two close sequences in D and ¢(f(zn)) converge to some complex value a in the w-plane. Since f(z) is normal, the cluster values of f(z) on {Zn} and {z;} are identical and thence, since ¢(§) is continuous, ¢(f(z;)) « a. LEMMA 3.2. _§ u(z) ppp_v(z) are real valued functions lp_ D, pppp f(z) = exp(u(z)+iv(z)) normal lp D implies u(z) lp normal lp_D. REMARK 3.3. An equivalent formulation of lemma 3.2 is that f(z) normal implies log‘f(z)‘ is normal. Proof pl Lemma 3.2. The mappings ¢1 on the g = f(z) plane to the non-negative real axis (including +m) by ¢1(§) = ‘g‘ and ¢2 on the non-negative real axis x = ‘g‘ to the closed real numbers by ¢2(x) = log x are continuous. Hence u(z) = ¢2(¢1(f(z))) is normal by lemma 3.1. THEOREM 3.4. ll u(z) ppp_v(z) are real valued normal functions 12.D pppp_ppp£_v(z) lp_bounded, then f(z) = exp(u(z)+iv(z)) lp normal lp_D. l6 l7 EIEEER Since exp x is a continuous function whose domain is the real line, exp(u(z)) is normal. If exp(iv(z)) is normal, then f(z) = exp(u(z))exp(iv(z)) is normal by corollary 2.8. Let {Zn} and {2;} be two close sequences in D and exp(iv(zn)) converge to a complex value a. Let I denote the cluster values of {v(zn)] and I' the cluster values of [v(z;)]. If x is in I, then exp(ix) = a. Since v(z) is normal, I = I', and thus exp(iv(z;)) d a by x in 1' implies exp(ix) = a. Therefore exp(iv(z)) is normal and the theorem is proved. The following example shows that the condition v(z) be bounded is needed. EXAMPLE 3.5. There exist u(z) and v(z) normal real valued such that f(z) = exp(u(z)+iv(z)) lp not normal. Let v(z) = Re(1/(1-z)) for z in D. Then v(z) is harmonic and normal l-l/2n n and since v(z) > 1/2 for all z in D. Let 2n 2; = l-l/(2n+fi)fl, O < n < 2. Then v(zn) = 2nn a +m, v(zé) = (2n+fl)n a +m, and by a simple computation p(zn,z;) = (1/2)log(4nn+2n-l)/(4nn-l) ~ 0 as n a a independent of n. If u(z) = 1, then f(z) = exp(u(z)+iv(z)) is such that f(zn) = e and, for H = 1/2, f(zé) = ie. Therefore f(z) is not normal even though u(z) and v(z) are normal and harmonic in D. The condition on the boundedness may be drOpped if we are given some other appropriate information. Note we don't even require v(z) to be normal. THEOREM 3.6. lpp u(z)‘pp‘p real harmonic function lp_D ppp v(z) its harmonic copjugate. lppp f(z) = exp(u(z)+iv(z)) lp normal lp_D lg and only lg u(z) lp_norma1 lp D. 18 ‘gpppg. The "if" part was done by Lappan in [14, p. 110] and the "only if" is lemma 3.2. The following lemma is an obvious corollary to theorem 2.1 which we will refer to in the discussion on logarithms of normal functions. LEMMA 3.7. ._£ u(z) and v(z) are real valued normal functions lp D, then f(z) u(z)+iv(z) lp also normal lp D. THEOREM 3.8. f f(z) is p normal function lp.D such that log f(z) lp_p well defined, single valued, continuous function lp D, pppp log f(z) lp_normal lp_D. 232223 Let {2“} and {zé} be close sequences in D and 10g f(zn) a a, a a complex number. Taking subsequences if necessary, we assume f(zn) converges to some complex value a. If a = 0 or a, then a»=‘jp+ib = a and log f(zé) 2.: also since f(zg) a a by f(z) normal. Hence, without loss of generality, we assume neither f(zn) nor f(zé) tend to zero or infinity; so arg f(zn) and arg f(zé) are defined for every n sufficiently large as well as any cluster values of {arg f(zn)] or {arg f(z;)}. Assume log f(zé) tends to some complex value 5. Then Rea = Rea since log‘f(z)‘ is normal (lemma 3.2); and arg f(zn) a Ima, arg f(zg) ~ ImB. If ‘Ima‘ = ‘ImB‘ = m, then a = B = a; so assume one is finite, say ‘Ima‘ < m. Connect zn and z; by the hyperbolic straight line Fn; the hyperbolic length of Fn’ 1(Fn), equals p(zn,z;) which converges to zero as n a o. Since f(zn) a a # 0 and f(z) is normal, there exists an N such that n12 N implies: If z is on Tn, then ‘f(z)-a‘ < ‘a‘/2 as well as ‘f(z£)-a‘ < ‘a‘/2. As z on Tn, n 2 N, travels from zn to z], arg f(z) changes continuously from l9 arg f(zn) to arg f(zé). Since ‘f(z)-a‘ < ‘a‘/2, ‘arg f(z)-arg f(zn)‘ < n/2 for any z on Tn. Since f(zé) a a # 0 also, arg f(zé) a Ima. So a = B and log f(z) is normal. LEMMA 3.9. f f(z) ppp p normal logarithm with arg f(z) bounded lp_D, then log‘f(z)‘ and ‘f(z)‘ are normal lp_D. ‘gpppg. Let {2n} and {2]} be close sequences and log‘f(zn)‘ d a, a a real number. Case 1. ‘q‘ = a. If a = +m, then Re log f(zn) > M, any preassigned real number M, for all n greater than some positive integer N. Hence Re log f(zé) a m as n a m, i.e. log‘f(z$)‘ a +n. The proof is similar if a = -m. Case 2. ‘a‘ < n. Then assume log f(zn) a a+iB # m since arg f(z) is bounded. By log f(z) being normal, log f(zg) d a+iB and so log‘f(z&)‘ « a. Therefore log‘f(z)‘ is normal. But log‘f(z)‘ normal implies ‘f(z)‘ normal by lemma 3.1. THEOREM 3.10. _£ f(z) has p normal logarithm with normal ‘ arg f(z) bounded lp D, pppp f(z) lp normal lp D. 2322:, The theorem is just a corollary of lemma 3.9 and theorem 3.4. The function f(z) = ‘2+z‘exp(iRe(l/(l-z))) is not normal in D but log f(z) = log‘2+z‘+iRe(l/(l-z)) is normal in D, from example 3.5 and lemma 3.7. Hence theorem 3.10 needs the hypothesis that arg f(z) be bounded. Let us now consider some applications of using exponentials and logarithms in the study of real and complex valued normal functions. 20 IEMMA 3.13. Let u(z) pp‘p harmonic real valued function lp_D and let there exist two close sequences, {2“} and {2;}, such ‘pppp u(zn) a a, u(zé) a B, a ppp B unequal real numbers, pp_n a n. Then for each real value (lncluding‘+n'ppp 4»), there exists p sequence [2:] plppp_£p_p_subsequence {zn } p£_{zn} pppp_£ppp u(z:)-°6§_§_k-oco. k REMARK 3.14. As is evident from the reference to be cited in the proof of lemma 3.13, we can require {zi} to be Such that u(zi) = 6 except for 6 = +n or «a. REMARK 3.15. Calling the above divergence property (Dh), we have the following characterization of not normal real harmonic functions: u(z) has prgperpyfi(Dh) 12.D.l£ and only lg u(z) lp not normal lp D. We will elaborate on a variant of this in chapter five. lgpppf‘pg‘lpppp 3.13. Let v(z) be a harmonic conjugate to u(z) and F(z) = exp(u(z)+iv(z)). F(z) is a holomorphic function in D such that ‘F(zn)‘ d exp a and ‘F(z;)‘ 4 exp B, exp a # exp B. So there are close subsequences {Zn ] and [2; ] of [Zn] and {2;} P P . * i: * * respectively where F(zn ) a a , F(z; ) a B , and a # B . By P P Lappan [12, p. 44] there is a sequence {wi} such that p(w16(,zn ) .. 0 and F(wi) - exp 6 (exp(-+oo) = +1», exp(-oe) = 0). So Pk u(wi) --0 6. With lemma 3.13 we may prove a theorem with real harmonic functions similar to theorem 2.10. THEOREM 3.16. Let u(z), v(z), w(z) pp real harmonic functions lp,D such that v(z) and w(z) are normal, and let K1 and K {pp constants such that v(z)+K1 S u(z) S w(z)+K2 for each z‘lp 2 21 D. ll lipppnf v(zn) > «n for each sequence {2n} pl p01nts lp_D for which w(zn) a +m, then u(z) lp_normal EE.D- Proof. Assume u(z) is not normal in D. Then there exists a sequence {2n} such that u(z) has property (Dh) on {Zn}. Then {Zn} has a subsequence {2n } for which there exist sequences l H - . l {znk] and [zn ] both close to {Zn ], satisfying u(zn ) a «a, k k u(zg ) a +m. But then lim inf v(z' ) = «n and w(z" ) a +m. k k—ee “k “k Since v(z) and w(z) are normal, this means that lim inf v(zn ) = -m k—Ioo k and w(zn ) a +n in violation of the hypotheses. Thus there is no k sequence {zn} on which u(z) has property (Dh), and the theorem is proved. Theorem 3.6 above shows the equivalence of the normality of the sum of normal harmonic functions with the normality of the product of non zero normal holomorphic functions. To this end we prove the following theorem on the sum of two normal harmonic functions. THEOREM 3.17. lpp u1(z) ppp u2(z) pp normal harmonic K functions lp D. ‘lf there are K (Ki > 1) such that 1’ 2’ ‘ui(z)‘2 S Ki‘u1(z)+u2(z)‘2, (i=1, 2), for every z lp D, then u1(z)+u2(z) lp normal 12,”- lEMMA 3.18. Let v(z) pp_p_harmonic conjugate Ep.u(z) ppp F(z) = exp(u(z)+iv(z)). Then lung,‘ (ux(z))2+(uy(z))2 p(F(2)) = 1+‘F(z)‘2 1+|u(z)|2 1] 2 :3. L113); "4:22;... _ 1...- 22 .2322£.2£.EEEEE 3.18. This computational fact follows from the fact that a2 S 2(cosh a)-l. We will also use the following theorems. THEOREM 3.A. (Lehto and Virtanen [15, p. 56]). lg f(z) ‘lp p meromorphic function lp_D, then f(z) lpmp normal function ll and only ll there exists p positive number G such that ‘f'SZZ‘ S G 1+‘f(z)‘2 1- z‘ 2 for each z lp D. THEOREM 3.B. (Lappan [13, p. 157]). If u(z) is'p harmonic function.lp D, then u(z) lp‘p normal function l: and only l: there exists p positive number G such that \flx>2+>2 G 1+‘u(z)‘2 S l-‘z‘2 for each 2 lp_D. Proof p£_Theorem 3.17. With F(z) = exp(u1(z)+u2(z)+iv(z)) where v(z) is a harmonic conjugate to u1(z)+u2(z), lemma 3.18 gives 2 2 ‘Fu(z)‘ S\j/((u1(z)+uz(z))x) +((u1(2)+u2(2))x) , 1+|P(z)|2 1+‘u1(z)+u2(z)‘2 Since \/(a+b)2+(c+d)2 s, /a2+c2 +fnz+d2 for a, b, c, d real, 2 2 2 2 43%”! S (u1(z)x) +(u1(z)y) +(u2(z)x) +(u2(z)y) . 1+‘F(z)‘2 1+|ul+u,(z>|2 Then, it is easy to check that for Ki > 1, K. 1 S 1 2 2 (i=1, 2). 1+‘u1(z)+u2(z)‘ 1+‘ui(z)‘ .;.'1.L.Lm§iA’ h‘ in. mains—iiimwwmm’mh “kugmza, 23 Since u1(z) and u2(z) are normal, there exist G and G2 such that l M S (RIGIfiZGZ) ___1_2_ 1+‘F(z)‘ 1-‘z‘ for all z in D. Hence F(z) is normal and theorem 3.6 implies u1(z)+u2(z) is normal in D. REMARK 3.19. By prOper modification of the above proof we could Show: If ui(z), i=1, 2,...,n, are normal harmonic functions and if there exist Ki’ i=1, 2,...,n, Ki > 1, such that 2 II II ‘u.(z)‘ S K.‘ E u.(z)‘2, then 2 u.(z) is normal. Also, by using 1 l j=1 J j=1 j the Cauchy Riemann Differential Equations, with vi(z) a normal harmonic conjugate to ui(z) and ui(z) not necessarily normal for i in some subdet J of {1, 2,...,n], we can Show: If 2 n 2 2 n 2 ‘vi(z)‘ S Ki‘ 2 uj(z)‘ for i in J and ‘ui(z)‘ S Ki‘jE1 uj(z)‘ J=1 n for i not in J, then 2 u.(z) is normal. The above remgfk hints at a relationship between the normality of a harmonic function and the normality of its harmonic conjugate. The next few resultsgmake the relationship clearer. THEOREM 3.20. lpp u(z) ppp_v(z) pp harmonic conjugates such that ‘v(z)‘ 2 k‘u(z)‘, k‘p fixed positive constant, for each 2 $2.D- ll_u(z) lp_normal EE.D» then v(z) lp also normal lp_D. Proof. \Jévx(z))2+(vy(z))2 = ‘f'(z)‘ and ‘qékux(z))2+(kuy(z))2 = k‘f'(z)‘ where f(z) = u(z)+iv(z). Since ‘v(z)‘ 2 k‘u(z)‘, we have 1/(1+‘v(z)‘2) s l/(1+‘ku(z)‘2). There- ->2+>2 S /<1>2+<1>2 G fore «0’ S_——- l+v(z)2 k(l+‘ku(z)‘2 k(1-‘z‘2) 24 for each z in D, some constant G, since ku(z) is normal and because of theorem 3.B. Hence v(z) is normal. THEOREM 3.21. ‘_£ u(z) ppp_v(z) are harmonic conjpgates lp D, ppp ll u(z) lp bounded lp_D, pppp_v(z) lp normal lp_D. ‘gpppp. Since u(z) is normal, F(z) = exp(u(z)+iv(z)) is normal. If v(z) is not normal, then there exist close sequences {zn} and {2;} in D such that v(zn) a 0, v(zé) ~ n as n a m from .1 . ' remark 3 5 Pick a subsequence {znk] of {Zn} so that [F(znk)} converges, say to the real number a. If a is so that 0 < ‘a‘ < a, then a = exp(B+i0) for some B (B = log‘a‘). But F(za ) a exp(B+in) # a, a contradiction of F(z) normal. Hence k a = 0 or m, or u(z ) fl -m or u(z ) a +m. Since this contradicts n n k k u(z) bounded, v(z) must be normal. COROLLARY 3.22. l£_f(z) has pp analytic logarithm lan with arg f(z) bounded, then f(z) lp normal lp_D. Ppppl. This corollary follows directly from theorems 3.21 and 3.10. EXAMPLE 3.23. There exist normal holomorphic functions with normal real pprt but not normal imaginary part. Let f(z) = 1/(1-z) = u(z)+iv(z) for z in D. Since f(D) = {w: Re w > 1/2}, f(z) and u(z) are normal in D. Consider the inverse images of Im w = 0 and Im.w = 1/2 from the range of f(z) for z in D. The first is the interval (-l,l) while the second is the circle of radius one, center 1+i, intersect D. Since the two inverse images are tangent, there exists a sequence {Zn} contained in the first inverse image and a sequence {2;} COntained in the second inverse image which are close but 25 v(zn) = O, v(zg) = 1/2 for all n. Hence v(z) is not normal in D while its conjugate u(z) is normal. One could also obtain theorems concerning the boundary values of normal harmonic functions of somewhat similar nature as Lehto and Virtanen in [15, pp. 58-62] and Vfiisfilfi in [17, pp. 29-30] employing theorem 3.6, lemma 3.18, and the results of Lehto and Virtanen. IV. VERY NORMAL FUNCTIONS DEFINITION 4.1. A complex, finite valued function f(z) in D is called very normal if there exists a positive number M such that, for each pair of points 21 and 22 in D, ‘f(zl)-f(zz)‘ < M p(zl,22). THEOREM 4.2. ‘_£ f(z) lp_very normal lp D, then f(z) lp normal 2 D. gpppp, The theorem is obvious from the fact that x(a,b) S ‘a-b‘ and the definition of a normal function. THEOREM 4.3. f f(z) lpflp continuous, complex, finite valued function lp_D, then f(z) lp_very normal ll and only l; there exists prositive number K such that K fgz'2-fng S 1-‘2‘ z'-z M(f(z)) = lim sup z'dz. for each z lp_D. Proof. Necessity. Fix 2 in D and let P be such that ‘f(z')-f(z)‘ < P p(z',z) for each 2' in D. Since IZ'-ZI (4.1) p(z',z) = 2 (1-‘2‘ )(1+€(z'az)) where e(z',z) a 0 as 2' a 2, we have lf(z:)-f(z)l < 2 P for all ‘2 '2‘ (I-Izl )<1+e 0. Hence, as 2' -o z, 26 27 P P M(f(z)) S S l-‘z‘2 1-‘E‘ for each 2 in D. Sufficiency. Let f(z) be continuous and M(f(z)) S 1_Kz for each 2 in D. We have 2 (1-‘2‘2)M(f(z)) = lim sup ‘f(z')'f(z)‘ 1; E , 2 2 a2 and, from (4.1), (1“2‘2)M(f(2)) = lim Sup lsz')-f(;)‘ 2L~z ' p(z',z) From the hypothesis and K S -z§-' we get 1-‘2‘ 1-‘2‘2 ' . lim sup [f(z 2 f(2)‘ < 2K. Zl-PZ 9(2 ,2) Let 21 and 22 be points in D and let L be the hyperbolic geodesic between 21 and 22. For each 2 on L, there is a 6 = 6(2) > 0 such that p(2',2) < 6(2) implies ‘f(z')-f(z)‘ < 4K p(z',z). Since L is compact, there exists a finite positive integer N and points 2i = 21, 2i,...,z§ = 22, 2; on L for i=1, 2,...,N, such that ‘f(zi)-f(z£_1)‘ < 4K p(z£,zi_1), i=2,...,N. Since L is a geodesic, we find ‘f(dl)-f(22)‘ < 4K 9(21,22). Hence f(z) is very normal in D. COROLLARY 4.4. _£ f(z) lp_holomorphic lp D, pppp_f(z)‘lp .1251 normal ll ppp pply lg there exists p positive number A.f pppp ties. Af ‘f'(z)‘ < IjTET séZZEZ each 2 in D. ‘—— fl' Ewdfia' I‘ITWWW‘ 7‘15“?“ b “4'. 7'11 . 'jc-‘ a. . . .1. 28 .EEQEE' Since f(z) is holomorphic in D, ‘f'(z)‘ = M(f(z)). From Lappan [12, p. 47] we see in the case of holomorphic functions that the class of uniformly normal functions and the class of very normal functions are the same. We now restate Lappan's definition of a uniformly normal function, dropping the requirement that the function be holomorphic. DEFINITION 4.5. A complex, finite valued function f(z) in D is said to be uniformly normal if, for each M > 0, there exists a finite number K > 0 such that for each 20 in D, p(2,20) < M implies ‘f(2)-f(zo)‘ < K. THEOREM 4.6. _f_f(2) lp very normal lp_D, then f(z) lp uniformly normal lp_D. Proof. Let f(z) be very normal and M, a positive constant, given. If 21 and z2 are such that p(2 2) < M, then 1,2 ‘f(zl)—f(22)‘ < PM where P is the constant of definition 4.1. Letting K = PM for the K of definition 4.5, we see that f(z) is uniformly normal. Since any bounded discontinuous function in D is uniformly normal but neither normal nor very normal, the converse of Theorem 4.6 certainly in not valid. Moreover, the function f(z) = min(1,\/E(;:O)) defined in D is normal, uniformly normal, but not very normal since lim sup z~0 lining-91 = .. COROLLARY 4.7. ‘lf u(z) lp_p_real harmonic function lp D, ,522251 u(z) lp very normal ll and only ll there exists p positive 2.52% K such that L g ,' 91-;1233' 1.vv,.", . Vuqul ‘ b 'r‘ ..13 29 >‘°'+)z+2+>2+(vy2 s 14ng for each 2 lp D. Proof. The sufficiency follows from M(f(z)) S 2 2 \Jéux(z)) +(uy(z)) +(vx(z))2+(vy(z))2 . For necessity, let f(z) = u(z)+iv(z) be very normal in D. By theorem 4.8, both u(z) and v(z) are very normal in D and hence by corollary 4.7 there are constants Ku and Kv so that (ux(2))2+(uy(z))2+(vx(2))2+(vy(z))2 s F 2+ (>>2+ <)>2+ (n2 Kfl" ‘\(ux(z)) (Uy Z ~ (Vx z (vy z s I:T;[ . The proof is completed by setting K = Ku+Kv. L 1-Wmim ti. 1:121er w 2-. “sum-.1 .' 3311‘“- '30 THEOREM 4.10. Let f(z) = u(z)+iv(z) pp holomorphic lp_D. The following are equivalent: (1.) f(z) lp vepy normal (2.) u(z) lp_very normal (3.) v(z) lp very normal. Proof. Since f(z) is holomorphic, we have that 5.2-3.3111 ‘ ‘f'(z)‘ =\/(ux(z))2+(uy(z))2 = (vx(2))2+(vy(z))2, and the theorem follows directly from corollaries 4.7 and 4.4. ..n " I‘ n‘_ In theorem 4.10 it is clear that one can't replace the word7 nw—fimwwaww-w-wwm 1"] ‘ "" v3; "holomorphic" by "harmonic". THEOREM 4.11. lg log f(z) lp very normal lp_D, then f(z) ‘lp normal lp D. IEEEER Let log f(z) be very normal in D and {Zn} and [2;] close sequences in D where f(zn) converges to some complex value a. Since log f(z) = log ‘f(2)‘+l arg f(z), where arg f(0) is fixed and arg f(z) continuously defined, theorems 4.8 and 4.2 imply that log ‘f(z)‘ and arg f(z) are both very normal and normal. If a is zero or infinity then log ‘f(zn)‘ tends to «n or +u respectively. Since log ‘f(z)‘ is normal, log ‘f(z;)‘ tends to «a or +2 respec- tively and hence f(zé) converges to 0 also. Therefore we may assume 0 < ‘a‘ < 0. Since log ‘f(z)‘ is normal, log ‘f(z$)‘ con- verges to log ‘a‘ and hence ‘f(zé)‘ a a. Also, from arg f(z) being very normal, for every 6 positive there exists a positive integer N(e) such that n >'N implies ‘arg f(zn) - arg f(zg)‘ < 6- Then as n tends to infinity, {‘f(2n)‘] and {‘f(2;)‘} have the same cluster values and so do {arg f(zn)] and {arg f(zé)] have the same cluster values. Thus f(zg) d a also and f(z) is normal. V. (A) SEQUENCES FOR NOT NORMAL FUNCTIONS In this chapter finite valued functions f(z) with range in the plane are considered; f(z) could be real valued. Unless other- wise specified, f(2) is harmonic means f(z) = u(z)+iv(z) and u(z) and v(z) are real valued harmonic functions. Most of the early material of this chapter can be generalized to functions with range on the Riemann sphere with chordal metric replacing absolute values. DEFINITION 5.1. Let [2n] be a sequence in D. Then {2“} is an (A) sequence for f(z) if there exists a sequence {2;} close to {an} such that lim sup ‘f(zn)-f(zr'1)‘ 2 n n-m LEMMA 5.2. Let w = f(z) pp harmonic lp_D and {Zn} p sequence lp,D such that lim f(zn) = a. ‘ll there lp.p sequence n—m {2;} close pp [2“] such that ‘f(z;)-a‘ > b (pp"f(2;)‘ < 1/b ll a = m) for some fixed b >'0, then {2n} lp_pp_(A) sequence for f(z). Proof. If a # m, then either u(z;)#L Rea or v(zg) ¥9 Ima. Without loss of generality we assume that u(zé) #H Rea. Then lemma 3.13 implies that there exists a sequence {2; } close to k [2n }, where {Zak} is a subsequence of {2“}, such that u(zgk) » +n. Hence {2n} is an (A) sequence for f(z). If a =‘c, the result follows immediately from the definition. THEOREM 5.3. If f(z) lp_harmonic and not normal lp_D, then there exists pp (A) sequence for f(z). 31 WVWH“ nag»; W‘fiii -’ ‘ ,uicai it. v1.1.3.1. ‘ a . O 32 Proof. Since f(z) is harmonic and not normal, remark 3.15 implies the hypotheses of lemma 5.2 are fullfilled for some sequence {Zn} In D. THEOREM 5.4. Let f(z) pp pp arbitrary function lp’D and {2n} p sequence pp_D pp which f(z) converges £2_a # m. .l£ there . .0112»! exists p sequence {2;} close pp_{zn} such that y '2' “at 2.. lim sup ‘f(2é)-a‘ > 0, then f(z) lp not normal. Ham Proof. The theorem follows immediately from the defini- ..o tion of a normal function. V w-emmn 2:! $‘f‘9‘M 3., 3,4,: -. . , F COROLLARY 5.5. Let f(z) pp harmonic lp_D. Then f(z) lp not normal l: and only lg there exists pp (A) sequence for f(z), denoted_py.{zn}, pppp‘pppp f(z) lp_bounded pp {2n}. ‘Ppppg. The "if" part is a corollary of theorem 5.4 and the "only if" part is a corollary to theorem 5.3 and the fact that f(z) is not normal. EXAMPLE 5.6. The boundedness pl_f(z) 92-{zn} lp theorem 5.4 pp_needed, even lg f(z) lp real valued and harmonic lp D. For example, let f(z) = Re(l/(l-2)), 2n = 1-exp(-n), and 9 2n = 1-anexp(-n) where 0 < an < an < 1, an a l, and +1 (l-an)exp(n) a a as n a a. Then g(zn,zé) = (1/2)log((2-anexp(-n))/(2an-anexp(-n))) « 0 as n a m I so that {2n} and {2“} are close sequences. Then ‘f(2n)-f(z;)‘ = (1-an)exp(n) ~.o as n d 9. Hence {2n} is an (A) sequence for f(z), but f(z) is normal since it is real valued, harmonic, and bounded below. 'THEOREM 5.7. Let f(z) = u(z)+iv(z) pp‘p harmonic function ‘23 D and bounded pp the sequence {2n}. Then {2n} lp_pp_(A) 33 sequence for f(z) ll and only l: <1-I znl 2) (“x (2,) > 2+(U1(Zn)) 2+6, (2,) > 2+|2 (l-‘zn‘2) (ux(zn))2+(uy(2n))2+-(vx(2n))2+-(vy(zn))2 lim sup 2 naa l+‘u(zn)+iv(zn)‘ Now assume {2“} is not an (A) sequence for f(z). With {gn(z)} as above, there exists an s so that {gn(z)] is bounded on D0 = [2: ‘2‘ S 3}. Hence {gn(z)] is a normal family in Do. So, as before, . fluenmnx)2+<(Regn(o>>y>2+<2+<<1msn<0>>y>2 0 > 11m sup nan 1+‘gn(0)‘2 (1‘an 2) (ux (Zn))2+(uy (2,1))2+2+>2+>2 K 1+‘u(2)+-iv(2)‘2 1-‘2 fp£_pppp_z lp D. DEFINITION 5.9. A function f(z) defined in D is said to be a R-normal function if the family F = {f(S(2))}, where 2' = S(z) is an arbitrary one-one conformal map of D onto D, has the property that for every sequence {fn}, fn in F, there exists a subsequence which either converges uniformly on every compact subset of D, or else converges uniformly to infinity on every compact subset of D. Every normal function that is harmonic, holomorphic, or bounded is R-normal and every R-normal function is normal. See Rung [16, p. 14] for an example of a normal function which is not R-normal. DEFINITION 5.10. Two sequences {2n} and {2;} of points in D such that there exists a positive constant K where lim sup g(zn,z;) S K are called essentially close sequences. n43 DEFINITION 5.11. Let {2“} be a sequence in D. Then {2n} is an (AB) sequence for f(z) if there exists a sequence {2;} o o - ' = essentially close to {Zn} such that 11: SUP ‘f(Zn) f(zn)‘ “' w Naturally every (A) sequence for a function f(z) is also an (AE) sequence for f(z). THEOREM.5.12. Let f(z) pp pp arbitrary function lp_D. lé there exists pp (AE) sequence for f(z), denoted py_{zn}, with N 35 f(z) bounded pp {2n}, then f(z) lp not R-normal. Proof. Let {2“} be the (AE) sequence, with {2;} the essentially close sequence such that lim sup ‘f(zn)-f(z;)‘ ==a flaw and let K be the constant in the definition of essentially close sequence. Without loss of generality we assume f(zn) a 0 and f(zé) « m. Consider the sequence {gn(z')}, gn(z') = f(Sn(z')), where Sn(2') = (2'+zn)/(1+2nz'). Let D0 = {2: ‘2" S tanh(K+1)]. Since gn(0) a 0 and g(S;1(zg)) a.” with 8;1(z£) in D0 for all n, the sequence {gn(z')] contains no infinite subsequence which con- verges uniformly to a finite valued function or to infinity on DO. Therefore f(z) is not R-normal. Let Fi(i=1,2) be simple continuous curves 21 = zi(t) (o s t < 1) in D such that ‘zi(t)‘ 2 1 as t 2 1. The 2932 Euclidean Frechet distance between f1 and F2 is .D(F1,F2) = max( lim su g(zl,F2), lim su p(22,F1)). zlerl ”1‘3”1 zzerz [22‘3”1 COROLLARY 5.13. lEE-Fl 339,?2 pp simple continuous curves it} D tending _tp the boundary such that 361,172) i_S_ finite. l_f_ f(z) _ip R-normal E D ppcl f(z) -. on Q 1‘1, ph_e_n_ f(z) an flrz. Ppppg. If {2i} is any sequence onI‘2 tending to the boundary of D, there exists a sequence {2:} on F1 and a finite constant K such that p(z:,z:) < K for all n sufficiently large. Since f(zi) S,» and by theorem 5.12, we havef(z§) a m. COROLLARY 5.14. ]lpp f(z) ppmp R-normal function lp D. l£.T1 tends non-tapgentiallyfipp_exp(ie) pp'C EIER f(z) a a pp F 1, then exp(ie) lp p Fatou point pl f(z) with p Fatou value a. ITT"“"“FISW"WFWVP Euann' 36 COROLLARY 5.15. Let f(z) ppnp R-normal function lp_D. l£_T1 tends non-tangentially pp_exp(ie) pp_C glpp f(z) bounded pp F1, pppp f(z) lp_bounded lp every Stolz angle pp exp(ie). Corollaries 5.13, 5.14, and 5.15 are analogues or gen- eralizations to Theorem 3, Bagemihl and Seidel [5]; Theorem 1, Rung [16]; and Theorem 4, Bagemihl [3], respectively. THEOREM 5.16. l£_f(z) lp_defined lp_D and there exists .pp (AE) sequence for f(z), pppp f(z) lp neither uniformly normal nor very normal. EEEEE' The theorem follows immediately from definitions 4.5, 4.1, 5.11, and theorem 4.6. THEOREM 5.17. If f(z) lp_not uniformly normal lp_D, pppp there exists pp (AE) sequence for f(z). Proof. Let f(z) be not uniformly normal. Then there exists K, 0 < K.< 9, such that for each positive integer n one . I . I Cl _ I . can find 2n and zn in D where p(zn,zn) < K an ‘f(2n) f(zn)‘ > n Hence {2“} is an (AE) sequence for f(z). Naturally then theorems 5.16 and 5.17 yield the following: COROLLARY 5.18. Let f(z) pp defined lp D. Then f(z) lp not uniformly normal ligand only l£_there exists pp (AE) sequence .gpp f(z). Following the lead of Lange [10] and Gauthier [8], but with cluster sets instead of range sets, we easily obtain some results concerning the cluster sets along hyperbolic disks with centers on (A) sequences. For a sequence {2“}, define the sequence of disks {A:} by A: = A:(zn) = {2: p(2,zn) < s} for each positive integer n. I 1 I imamnmflwrenvamnwr ”l.‘ _ ,g."2.“1£11 3: 3111111 1 I h 37 THEOREM 5.19. f {2n} _i_s_fli_ (A) sequence for f(z) = u(z)+iv(z) harmonic lp_D with f(z) bounded pp {2n}, and Q a: for each 9 > 0 letting C(f, U A:(zn)) = C(f,A:) = n (U f(A:(zn)) ). i=1 j=l n=j then C(f,A:)‘lp either p line parallel pp one pg the axes for each a sufficiently small pp for every real number x, there are w1, w2 ,lp C(f,A:) such that Re w1 = x and Imw2 = x. EEQEE- Consider C(u,A:) and C(v,A:). If either one, say C(u,A:), is a singleton for some 60’ then C(v,A:) will at least be a closed half line for every 6 S so since f(z) is continuous and {2“} is an (A) sequence for f(z). Lemma 3.13 and remark 3.15 how- ever imply v(z) is not normal and C(v,A:) is the whole (closed) real line. Therefore we have the first situation of the theorem's conclusion. The only other situation is if for every 6 > 0, neither C(u,A:) nor C(v,A:) is a singleton. We know there exists a sub- sequence {zn-] where f(zn ) a B = a+ib%m from hypotheses. From k k the-"neither nor" statement there exist {2; ] and {2" ] close k “k 1 l to {znk }, a subsequence of {znk], where u(zt'lk ) a c # a and I 1 v(zg ) a d # b. Hence neither u(z) nor v(z) is normal and the k 1 conclusion follows. EXAMPLE 5.20. There exists p harmonic function f(z) which .lp not normal and, for some (A) sequence {2n} for f(z) and each s, 0.< e < a, C(f,A:) properly contains jpst one line and pp_lp not the whole plane. Let f(z) = Re(z)+iRe(exp(2)) with the upper half plane H = {2: z = x+iy, x,y real, y > 0] as its domain and 38 * p (21,22) as the hyperbolic distance between 21 and 22 in H. Let 1'1 to the real axis. Then, as {2“} on F1 and {2;} on F2 go to infinity * such that Imzn = Imzé, we have p (2n,z;) a 0 and {f(zn)], {f(z;)] have bounded disjoint cluster sets. Hence f(z) is not normal. From the hyperbolic metric in H and Re(f(x+iy)) = x and using the natural definition of an (A) sequence in H we see that {2n = 2nni] is an (A) sequence for f(z); i.e. C(x,A:) is the whole closed real line for any a > 0. Let wO be in C(f,A:). This implies there exists a sequence {2'} z' = z'+iy' in A6 such that x' a Re w . n ’ n n n n’ n 0 Since -1 S cos y; S l, {Re(exp(2é))] has a cluster set contained in {w: w b exp(Re wo), -l S b S 1]. Hence Im wO obeys the in- equalities -exp(Re wo) S Im w S exp(Re wo). One also could easily choose ya so that Im W0 is any number in the closed interval from -exp(Re wo) t0 exp(Re W0). Hence {w: w = a+i0, a real] is properly contained in C(f,A:) = {w: w = a+ib exp(a), a real, -1 S b S l} and this is properly contained in the closed plane. Clearly only one line is contained in C(f,A:). There are many such functions with the properties of example 5.20. For example, any function of the form N fcr+iy) = x +-i Z exp(c,x)(a.cos c.y +-b,sin c.y), where N is a 3:1 J J J J J positive integer, aj and bj real and not all zero, and cj rational with at least one non-zero, also satisfies example 5.20. THEOREM 5.21. lp£_f(z) pp harmonic and not normal EB.D- Then, for each (A) sequence {2n} fpp f(z) plpp f(z) bounded pp {2n}, either C(f,A:)‘lp‘p line for a sufficiently small pp and F2 be two distinct rays in the upper half plane perpendicular WV 21 _yu'ab 1 '1 - 7 5 Ital— =52: fish yuan ‘J ‘wfimnfivafiflwfiwp, '2'. 39 C(f,A:) intersects every half plane. Proof. The theorem is clear from the fact that exp(ie)f(z) is harmonic and not normal for all real 9. [‘EJ; 41. 511111111,. . h BIBLIOGRAPHY 10. 11. 12. 13. BIBLIOGRAPHY L. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1953. F. Bagemihl, Some Boundary Properties of Normal Functions Bounded on Nontangential Arcs, Archiv der Mathematik, XIV, (1963), 399-406. F. Bagemihl and W. Seidel, Sequential and Continuous Limits of Meromorphic Functions, Ann. Acad. Scient. Fenn A. I., 280, (1960), 1-16. , Behavior of Meromorphic Functions on Boundary Paths, with Applications to Normal Functions, Archiv der Mathematik, XI, (1960), 263-269. L. Bers, Theory of Pseudoanalytic Functions, New York University, 1953. (Mimeographed Lecture Notes). , Local Theory of Pseudoanalytic Functions, Lectures on Functions of a Complex Variable, ed. Wilfred Kaplan et al. (Ann Arbor: The University of Michigan Press, 1955). J. Cima, A Nonnormal Blaschke Quotient, Pac. J. Math., 15, (1965), 767-773. P. 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