WI 1 1 l WWW M \ NQRMAL LIGfiT ENTEREQR FUNCTiQNS DEFINED EN THE UNET DiSK Thesis for the Degree of Ph.. D. MiCHEGAN STATE UMVERSITY JUHN HENRY MATHEWS 1969 ;’ all!" . 4,, _, .- Lu..-'m-'- - fins I LIBR .‘l 527 Y Michigan 313” Universxty THFSis This is to certify that the thesis entitled NORMAL LIGHT INTERIOR FUNCTIONS DEFINED IN THE UNIT DISK presented by John Henry Mathews has been accepted towards fulfillment of the requirements for Ph.D. degree in Maghemagics MflLzQemfi Major professor Date%‘9—z )3) /76? 0-169 M 3.“):— EH- s E. autumn av “‘ I "DAB & SONS' i nnnv nm in ‘m‘ ' ABSTRACT NORMAL LIGHT INTERIOR FUNCTIONS DEFINED IN THE UNIT DISK By John Henry Mathews Let f be a light interior function from the unit disk into the Riemann Sphere. Then f can be factored f = g o h where h is a homeomorphism and g is a meromorphic function. Although this factorization is not unique it is shown that there is a unique factorization type. Conditions are established to determine the normality of f; and it is shown that boundedness is not sufficient for a light interior function to be normal. Several examples are presented which show that the classical theorems of Fatou, Koebe, Lindelgf and Riesz cannot be extended for even bounded normal light interior functions in the unit disk. For example, there exists a bounded normal light interior function in the unit disk for which the total outer angular cluster set is one point. Conditions are established to determine when some of the classical theorems will hold for light interior functions. It is shown that several theorems hold for pseudo-meromorphic functions. For example, Koebe's theorem and Lindelgf's theorem remain true for normal pseudo-meromorphic functions. John Henry Mathews Let f be a light interior function in the unit disk with factorization f = g o h where h is a homeomorphism of the unit disk onto the unit disk and g is a non-constant meromorphic function in the unit disk. Then the asymptotic behavior of f is shown to be closely related to the asymptotic behavior of its component factors g and h. NORMAL LIGHT INTERIOR FUNCTIONS DEFINED IN THE UNIT DISK By John Henry Mathews A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1969 ACKNOWLEDGEMENT I wish to thank Professor Peter Lappan for suggesting the investigation of normal light interior functions and for his patient guidance throughout the preparation of this thesis. ii TABLE OF CONTENTS I. INTRODUCTION ...................................... 11. NORMALCY AND THE sroi'mw FACTORIZATION ....... l. StoIlow factorization 2. Uniqueness of the StoIlow factorization 3. Necessary conditions for both f and g normal 4. Sufficient conditions for f normal III. FAILURE OF THE CLASSICAL THEOREMS ................. l. Fatou's theorem and Koebe's theorem 2. Lindelgf's theorem 3. Riesz's theorem IV. GENERALIZATIONS AND APPLICATIONS TO K-EM FUNCTIONS . 1. Quasiconformal functions and pseudo-meromorphic functions 2. Normality 3. Preservation of Stolz domains 4. Preservation of Koebe arcs V . ASYMPJZOTIC BEHAVIOR ............................... BIBLIOGRAPHY ...................................... iii I. INTRODUCTION Lehto and Virtanen [15] defined the concept of a normal meromorphic function as follows: If g is meromorphic in a simply connected domain G, then g is normal if and only if the family {g(S(z))}, where S(z) denotes any arbitrary one-one conformal mapping of G onto G, is normal in the sense of Montel. Mero- morphic normal functions defined in the unit disk were found by Lappan [14] to be precisely those which are uniformly continuous with respect to the non-Euclidean hyperbolic metric in the unit disk and the chordal metric on the Riemann sphere. We will say that a function f mapping the unit disk D into the Riemann sphere W is a normal function in D if and only if f is uniformly continuous with respect to the non-Euclidean hyperbolic metric in D and the chordal metric in W. VHisglg proved [21, Theorem 2, p. 17] that if one uses the Lehto-Virtanen definition of normal then there are no non-constant normal meromorphic functions in the finite complex plane .0. Yosida [23, p. 227] has defined the concept of a normal meromorphic function in Q as follows: If g is a meromorphic function in 0, then g is normal if and only if the family {g(a + z): a E Q} is normal in the sense of Montel. Recently, Gauthier [10, p. 560] has proved, using Lappan's technique, that Yosida's definition is equivalent to the following: A meromorphic function g in Q is normal if and only if it is uniformly continuous with respect to the Euclidean metric in Q and the chordal metric in W. This definition does not exclude all non-constant meromorphic functions and includes, for example, elliptic functions, periodic functions and rational functions in. Q [23, p. 227]. We will define the concept of a normal function in Q as follows: A function f mapping the finite complex plane 0 into the Riemann sphere W is a normal function in_ 0 if and only if f is uniformly continuous with respect to the Euclidean metric in Q and the chordal metric in W. Normal meromorphic functions in D have been investigated by Lehto and Virtanen [15], Bagemihl [2], Bagemihl and Seidel [3], [4], Lappan [l3] and others. The question has been posed: To what extent do the results depend upon the fact that the functions are meromorphic? In this dissertation we investigate the behavior of normal light interior functions. A function f mapping a domain G into the Riemann sphere W is said to be light if for every point w E f(G) the set f-1(w) is totally disconnected, and f is said to be interior if for each open set U<: G the set f(U) is open in w [22]. The following definitions and conventions will be used. We shall denote by C the unit circle and by D the open unit disk in the finite complex plane 0. Let W denote the Riemann Sphere, and let x(w1,w2) represent the chordal distance between the points w1,w2 E W. If A and B are sets in W then x(A,B) denotes the chordal distance between the sets A and B. In the unit disk, let p(zl,22) denote the non-Euclidean hyperbolic distance between the points z 22 E D [5, Chapter 2], [12, Chapter 15]; 1, 9(zl,zz) = s1n<<1+u>/<1-u>> = tanh'1 where u = |(zl-zz>/(1-Z;22)|, alternately where F ranges over all paths joining 21 to 22. If {2n} and {2;} are two sequences of points in D with p(zn,z;) a O, we shall say that {2“} is close to {2;}, or that {2n} and {z'} are close sequences. n Let f be a function from D into W and let e19 be a point of C. We define the cluster set C(f,e) of f ‘25 e19 as follows: C(f,9) is the set of points w E W for which there exists a sequence {2n} of points in D with zn d e19 and f(zn) a w. Furthermore, the total cluster set C(f) of f is given by c(f) =[Jc(f,e), where the union is taken over all e 9 (O s e < 2n). If S is a subset of D and e19 6 [S.fl C], where S is the closure of S, we define the cluster set CS(f,e) f f at e19 relative to S as follows: CS(f,9) is the set of points w E W such that there exists a sequence {2n} of points in S with z a e16 and f(z ) a w. n n . 19 By a Stolz domain A 35 e- we mean a set of the form {2 e D: -n/2 < (251 < arga - 2/819) < (b2 < "/2}: and by a terminal Stolz domain pg e19 we mean a set of the form 16 A n {2: ‘2 - e l < e} (0 < e < 1)- . . . i The function f 18 said to have a Fatou p01nt at e 9 with Fatou value c, or angplar limit c, if f(z) a c as 2 a e19 from . . . 16 within each Stolz domain A at e . The outer angular cluster set cA(f,e) ,3: f ‘35 819 is defined as follows: CA(f,e> a: cA where the union is taken over all Stolz domains A at e19. Using this notation we see that e19 is a Fatou point with Fatou value c if and only if CA(f,9) = {c}. A simple continuous curve F: z(t) (O s t < 1) contained in D is called a boundary path if ‘z(t)| « l as t a l. The end of a boundary path 1‘ is the set F n c. If a boundary path i . F ends at e 6 then P is said to be a Jordan are at e19. A boundary path P: z(t) (O s t < 1) is an asymptotic path of f for the value c provided f(z(t)) a c as t a l. The point c is called an asymptotic value, or asymptotic limit, of f if there exists an asymptotic path of f for the value c, and c is said to be a point apymptotic limit of f if there exists an asymptotic path of f for the value c whose end consists of a single point. Let A be an open subarc of C, possibly C itself. A Koebe sequence of arcs, relative to A, is a sequence of Jordan arcs {Jn} in D such that: (a) for every 6 > 0, Jn<: {z E D: lz - a] < e, for some a E A} for all but finitely many n, and (b) every open sector A of D subtending an arc of C that lies strictly interior to A has the property that, for all but finitely many n, the arc Jn con- tains a subarc Ln lying wholly in A except for its two end points which lie on distinct sides of A. If f is a function in D and if c E W, we say that f has the limit c along the sequence of arcs {Jn}, provided that, for every 3 > O, x> < e for all but finitely many n. We will write f(Jn) d c. When {Jn} is a Koebe sequence of arcs we will call c a Koebe limit. II. NORMALCY AND THE S'ro'I'ww FACTORIZATION l. Stoilow factorization Let f be a light interior function from the unit disk D into the complex plane 0. Stoilow [20, p. 121] has shown that f has the representation f = g o h where h is a homeomorphism of D onto a Riemann surface R and g is a non-constant analytic function defined on R. Church [8, p. 86] pointed out that this result can be extended to light interior functions which map D into the Riemann sphere W provided g is allowed to be mero- morphic. In view of the uniformization theorem [1, p. 181] there exists a conformal mapping of R onto either D or 0. Therefore, if f is a light interior function from D into W then f has a StoIlow factorization f = g o h where h is a homeomorphism of D onto D (or Q) and g is a non-constant meromorphic function in D (or 0). Conversely, if h is a homeomorphism of D onto D (or Q) and g is a non-constant meromorphic function in D (or 0) then the function f = g o h is light interior. 2. Uniqueness of the Stoilow factorization DEFINITION 1. Let h 23 g homeomopphism pf D onto D (or 0). _f h .ig uniformly continuous with respect £9 the non- Euclidean hyperbolic metric 13 its domain D and the non-Euclidean hyperbolic metric 22 its range D (25 the Euclidean parabolic metric ‘13 its range 0), then wg_shall say that h 'ig HUC (25 PUC). DEFINITION 2. Let f ng_§Llight interior function 13’ D with StoIlow factorization f = g o h where h ‘ig'g homeomorphism 2i D onto D Log. 0) and g Eénon-constant meroLm-phic function _i_r_11 D (g Q). _I_f__ h .i_§_ HUC (pg PUC) then f has .g'type I factorization; otherwise f has §.type II factorization. THEOREM 1. f f .13 a light interior function i3 D then — f '_23 g unigpe factorization 2123, Proof. Case i. The light interior function f has a StoIlow factorization f = g o h with h a homeomorphism of D onto D. Suppose f also has the Stoilow factorization f = G o H where H is a homeomorphism of D onto D (or 0). Then as pointed out by Church [8, p. 88] h o H“1 is a conformal homeo- morphism, hence from Liouville's theorem h o H"1 must be a con- formal homeomorphism of D onto D. In view of Pick's theorem [12, Theorem 15.1.3, p. 239] both h o H“1 and h“1 o H are HUC. Since the composition of two uniformly continuous functions is uniformly continuous, it follows that h is HUC if and only if H is HUC. Eggs ii. The case when f has a Stoilow factorization f = g o h with h a homeomorphism of D onto 0 is handled similarly, and the proof of the theorem is complete. There is an abundance of HUC homeomorphisms, for example, every conformal homeomorphism of D onto D is HUC. The existence of a homeomorphism of D onto 0 which is PUC is established in the following theorem. THEOREM 2. There exists 2_homeomorphism h _f' D onto 0 we we. Proof. Define the mapping h in D by h(z) = z p(O,z). Then it is easy to verify that h is a homeomorphism of D onto 0. Let An = {z: n s p(0,z) S n+1]. Let n 2 3 be fixed but arbitrary; the proof will be complete if we can find a constant K, independent of n, such that |h(zl) - h(zz)‘ s K p(zl,22) for each pair of points 21, 22 E An with p(zl,zz) < 1. Let 21, 22 E An’ with p(zl,zz) a l, where n 2 3 is arbitrary 19 192 but fixed. We may assume that 21 = rle and 22 = rze with r1 s r2. Then we have the following inequalities |h - h<22>|= lzlp - 22p<0.22>| 192 e p(0,r1)| IA ‘zlp(0,zl) - r1 192 162 p(03r1) - r18 9(09r2)| + |r1e 19 2 + ‘rle p(0,r2) 22p(0,zz)‘ IA - + 2nl92 91|p(0,r1) p(zl,z2) + ‘rz - rllp(0,r2) + + . P1 P2 P3 Consider the first term P1. Let F be non-Euclidean 161 192 goedesic joining rle to rle . Let R be a real number (0 < R.< 1) for which p(O,R) = n-l. Then I62 - elI R S I Idzl l-R I. 1-|z 191 192 = p(rle arle ) S p(zlazz)~ Also, p(0,r1) s 2 p(O,R) so that we obtain IA I92 - ellp<0.r1> ((1 - R2)/R)p(zl,zz)2 p(o,R> (p(0.R><1 - R2>/R)2 o<21.22> Consider the third term P3. We observe that r r - r I r 2 2 l 1 dx s = . l - r1 r1 1 - x Also, p(0,r2) s 2 p(0,r1) so that we obtain IA 2 Ir2 - r1Ip(0,r2) ((1 - r1>/r1>p2 p(0,r1) = (p(0,r1)(1 - r§)/r1>2 “(21:22) IA 2 p(zl,zz). Finally, combining the estimates for P1, P2 and P3 we obtain |h(zl) - h(22)I 5 (4n + 3)p(zl,22). We choose K = (4n +-3) and the proof is complete. 10 3. Necessary conditions for both f and g normal Noshiro [18, p. 154] (or Yosida [23, p. 227]) has divided the class of normal meromorphic functions in D (or -0) into two kinds which are defined as follows: A normal meromorphic function g in D (or O) is of the first kind if the normal family {g(%—E—§;): a E D} (or the normal family {g(a + z): a E 0}) admits no constant limit; otherwise g is of the second kind. THEOREM 3. Let f 23 g_normal light interior function 23 D with Stoilow factorization f = g o h where h i§.2.h°me°' morphism pf D onto D (or Q) and g is.g non-constant mero- morphic fungtion‘ip_ D (p£_ O). f g ‘ig 3 normal meromorphic function i2 D (or Q), then h .i§_normal. Furthermore, if g £3 normal meggmorphic function pf the first kind ill D (g Q), Ehgp 'h .ifi HUG (pg PUC). 2E22£3 .ggpg i. The normal light interior function f has a StoIlow factorization f = g o h with h a homeomorphism of D onto D. If h is not normal there exist close sequences {Zn} and {2;} such that h(zn) a ehy and h(zé) a e1B with 0 < B - a < 2n [14]. For each integer n, let Jn be the non- u Euclidean geodesic joining zn to zn. Then {h(Jn)} is a sequence of Jordan arcs such that for every 3 > O, h(Jn)C{zED:l-e 0 such that p(h(zn),h(z;)) 2 6, and there exists a constant c such that f(zn) a e. Let h(zn) - z Sn(2) = 1 - h(zn) z and let Gn(z) = g(Sn(z)). Then the normal family {on} has a subsequence which converges uniformly on each compact subset of D to a meromorphic function G [15, p. 53]. Let Jn be the non-Euclidean geodesic joining zn to z; and let Ln = h(Jn). Then d(Ln) = d(S;1(Ln)) 2 6, where d(E) is the hyperbolic diameter of the set E<: D. From the normality of f we have f(Jn) a c, so that g(Ln) a c, and hence Gn(Sn1(Ln)) a c. For r (0 s r s 6) fixed, there exists a point Zn 6 S;1(Ln) such that p(0,zn) = r. Let 20 be a cluster point of the sequence {Zn} on the circle {2: p(O,z) = r}. Choosing a subsequence of {CH} if necessary, we can assume that Zn.a Z0 and Gn(zn) a c. A familiar argument (see e.g. [6, p. 179]) in the theory of continuous convergence shows that C(Z ) = c. Since r (O s r s 6) was arbitrary, 0 is a limit 0 12 point of values for which G assumes c and hence G s c in violation of our hypothesis. Therefore h is HUC and the proof of the first case is complete. §2§g_ii. The normal light interior function f has a StoIlow factorization f = g o h with h a homeomorphism of D onto 0. In this case h is always normal and when g is a normal meromorphic function of the first kind in 0 the proof is handled similarly to Case i; and the proof of the theorem is complete. 4. Sufficient conditions for f normal Every bounded holomorphic function is normal [3], but the following result shows that boundedness is not sufficient for a light interior function to be normal. THEOREM 4. f‘g homeomorphism h ‘pf D onto D lg not HUC, then there exists 3_Blaschke product B 33' D such that the bounded light interior function f = B o.h _i§ not normal 12' D. Proof. If h, is not HUC there exist close sequences I I {2“} and [2n] and a 5 > 0 such that p(h(zn),h(zn)) 2 6. Let h(z ) = w and h(z') = w'. Since h is uniformly con- n n n n tinuous on compact subsets we necessarily have that Ian a 1, I . Izé‘ a l, lwnI « l, and IwnI a 1. Hence, chOOSing a subsequence of {Wu} if necessary, we may assume that {Wu} is a Blaschke a: sequence, i.e. 2 (l - IwnI) < m. We now construct a Blaschke n l subsequence {wn ] of {Wu} and a corresponding subsequence k [w] ] of {wé}. k 13 = , = . = . I Let wn w1 and wn1 WI, and let r1 min{Iwn1I,Iwn I} and R1 = max [Iwn I,Iw;1|}. We can find an integer n2 > 1 such _ . -l 2 that for r2 - m1n{|wn2I,Iw;2|] we have p(R1,r2) 2 tanh (1-1/2 ). Let R2 = maxflwn I,Iw; I]. We proceed inductively to obtain 2 I I o subsequences [wnk] and {wn ] of {Wu} and [wn], respectively, such that p(Rk-l’rk) 2 tanh-1(1-l/k2) for each integer k 2 2, where r - min{Iw \,Iw' I] and R = max{‘w |,|w' I]. k nk nk k nk nk It follows easily that -l 2 , tanh (1-1/(k+l) ) (1 s kt< J) I p(wn ,wn ) 2 k j tanh-1(l-l/k2) (1 s j < k), and hence , 1 - 1/(k+1)2 (1 s k< j) wn - wn k j 2 l - wn w' 2 k nj 1 - l/k (1 s j.< k). Recall that p(wn ,w; ) 2 5 > O (k = 1,2,...) so that Set B(z) = H w k=1 n (l - w z) k nk Consider B(w; ) for j 2 l, l j l wn - “d wn - w; wn - w' - . . n. |B|=n k J. J 1.3; k 1 nj k=1 1 - wn w; 1 - wn w; k=j+l l - wn w; R J J' j R J -1 j‘1 2 “D 2 2 (tanh 5) n (1 - 1/(k+1) ) n (1 - 1/k) k=l k=j+l -l _ m = (tanh 1 6) H (l - l/kz) = £22%———§'> O. k=2 Let f = B o h. By assumption {2 ] and {z' ] are “k “k necessarily close sequences with II II lim f(znk) lim B(h(znk)) lim B(wnk) = O and lim f(z; ) lim B(h(z$ )) lim B(w; ) f O. k k k By a theorem of Lappan [14, Theorem 2, p. 156], f is not normal and the proof is complete. Let f be a light interior function in D with StoIlow factorization f = g o h where h is a homeomorphism of D onto D and g is a non-constant meromorphic function in D. The previous theorem suggests that the normality of g does not insure the normality of f. An even stronger statement is the following result. THEOREM 5. There exists 5 homeomorphism h f D onto D with the property: f g i§_§_non-constant normal meromorphic function ip_ D, which has two distinct asymptotic limits, then the light interior function f = g o h lg not normal ip_ D. Proof. Construct a sequence {Rn} of real numbers 0 = R.1 < R.2 <...< Rn.<°"< l for which p(Rn’Rn+l) = l/n. Define 15 the mapping h in D by _ is = . . _ h(z) h(re ) r exp(1e +'2n1(r Rn)/(Rn+1 Rn)) for Rn s r<< Rn+ (n = 1,2,...). It is easy to verify that h l is a homeomorphism of D onto D. Since g has two distinct asymptotic limits, a theorem of Lehto and Virtanen [15, Theorem 2, p. 53] implies that g has two distinct radial limits. Let Ta and TB be the radii which terminate at the points eh: and e16, reSpectively, for which g(rehy) a a and g(reia) « b with b # a. Now the radii of D are mapped onto spirals by h-l. Let _ -l 1) 2n and h (re) n [Knew p(zn,zé) s p(Rn,Rn+1) = l/n with f(zn) = g(h(zn)) a a and h-1(Ta) n [Rn,Rn+ ) = 2]. Then f(zé) = g(h(z;)) u b. Hence, by a theorem of Lappan [14], f is not normal and the theorem is proved. Since a bounded holomorphic function in D is normal and possesses uncountably many distinct radial limits [9] we obtain the following corollary. COROLLARY. There exists §”homeomorphism h g£_ D onto D with the property: f g ‘ig g non-constant bounded holomogphic function 32' D, then the bounded light interior function f = g o h .li not normal i3 D. We now determine conditions on h and g which insure the normality of f. Since the composition of two uniformly continuous functions is uniformly continuous the first result in this direction is obvious. 16 THEOREM 6. Let h bg_g,homeomorphism 9f_ D onto D (9.; 0) Mi; HUC (9;; PUC). If g _i_s_gnon-constant namelmsmmcraliia_a___fu ction in. D (at. 0), ___then __the _g_li ht inggzigrhfignggigp, f = g o h i§_norma1 12. D. Furthermore, if_ both, h and h"1 fire, HUC, then g I§.Dorma1 ip_ D if_and only if. f lineman D- DEFINITION 3. Let h be a homeomorphism p£_ D onto D. Define the set F(h) 2§_follows: e16 6 F(h) if there exist close sequences [2“] and {zé} and §_ 6 > O for which 16 I p(h(znzh(zn)) 2 6 and h(zn) a e THEOREM 7. Let h p£_2_normal homeomorphism.2£. D onto D. If. g $2.2.non-constant normal_meromorphic fupction in D which is continuous on D U F(h), thgn the light interior function f = g o h 12 normal in D. Proof. If f is not normal there exist close sequences I _. d I _. {2n} and {2n} such that f(zn) a an f(zn) b with b # a [14]. It follows from the normality of g that [h(zn)} and {h(z&)} are not close. Choosing a subsequence of {Zn} and a corresponding subsequence of [2;] if necessary, we may 16 ie 19 assume that h(zn) a e and h(zé) d 6 with e E F(h). But g is continuous on D U F(h) and hence = . I = . | = o = . = b 11m f(zn) lim g(h(zn)) 11m g(h(zn)) lim f(zn) a which is a contradiction. Therefore f is normal and the proof is complete. III. FAILURE OF THE CLASSICAL THEOREMS We now investigate the boundary behavior of normal light interior functions and show that the classical theorems cannot be extended even for bounded normal light interior functions. 1. Fatou's theorem and Koebe's theorem Fatou's theorem [9] states that a bounded holomorphic function in D possesses radial limits at almost every point of C. The following result shows that a bounded normal light interior function need not possess any point asymptotic limits. Koebe's theorem [19] states that a non-constant bounded holo- morphic function in D possesses no Koebe limits. The following result shows that a bounded normal light interior function can possess uncountably many distinct Koebe limits relative to C. THEOREM 8. There exists g_homeomorphism h pf. D onto D with the property: f g is a non-constant normal meromorphic function ip_ D, then the light interior function f = g o h ig normal and possesses pp point asymptotic limits. Furthermore, if_ g possesses 3 point asymptotic limit, then f possesses g Koebe limit relative £3 C. Since a bounded holomorphic function in D is normal and possesses uncountably many distinct radial limits we obtain the following corollary. l7 l8 COROLLARY. There exists-5 homeomorphism h f D onto D with the property: If g ‘i§.§ non-constant bounded holomorphic function i2, D, then the bounded light interior function f = g o h is pormgl and possesses no point asymptotic limits. Furthermore, f possessgs uncountably many distinct Koebe limits relative £2 C. Before proving Theorem 8 we establish the following lemma. LEMMA 1. There exists a homeomorphism h of D onto D *H‘am—n— 4‘. __ such that the radii pf. D a£p_mapped onto spirals and h ‘ip HUC. Proof. Construct a sequence {Rn} of real numbers, = ... ... 1 0 RO>.h>> s ponexpue1 + il’n(r1)),§n(r1)exp(iez + i‘I’n(r1))) + p(§n(r1)6XP(192 +>iwnmnexp> + p(§n(r1)eXP(192 +~iin>,¢nexp + . P1 P2 +P3 Consider the first term P1. From the fact that §n(r1) 3 r1 we obtain P1 = p(§n(r1)exp(iel),§n(r1)eXP(i92)) s p(rlexp> s p(zl,zz). Consider the second term P2. From the facts that 2 = l l - §n(r1) S Rn and 9(Rn:R ) /( Rn) we obtain n+1 20 Y (r ) n 2 4» (r1) de P2 3 n 2 Yn(r1) 1 - §n(r1) ZTT r s tp(Rn,r2) - p(Rn.rl)] 2 (1 - Rn)p(Rn’Rn+l) = 2n p(r1,r2) s 2n p(z1,zz). Consider the third term P3. .From the fact that p(§n(r1),§n(r2)) s p(r1,r2) we obtain P3 = P(§n(r1):§n(r2)) S p(rlsrz) S p(Zla22)o 1, P2 and P3 we obtain p(h(zl),h(z2)) s (2 + 2n)p(zl,22). We choose Finally, combining the estimates for P K = (2 + 2n) and the proof is complete. ‘Egggf pf Theorem 8. Let h be the homeomorphism of Lemma 1. Let g be a non-constant normal meromorphic function in D. Then by Theorem 6, the light interior function f = g o h is normal. If f has a point asymptotic limit c along a boundary path P, then it is easy to verify that h(F) is a spiral asymptotic path of g for the value c. Construct a Koebe sequence of arcs {Jn} in D be letting the Jn be the con- secutive turns of the spiral h(F). Then g(Jn) a c, and by Theorem 1 of [4], g E c in violation of our hypothesis. Therefore f has no point asymptotic limits. If g has a point asymptotic limit c along a boundary path A, then h-1(A) is a spiral asymptotic path of f for the 21 value c. Construct a Koebe sequence of arcs {Jn} in D by letting the Jn be the consecutive turns of the spiral h-1(A). Then f has the Koebe limit c relative to C and the proof is complete. 2. Lindelgf's theorem Lindele's theorem [9] states that if a bounded holo- morphic function f in D possesses the point asymptotic limit c at e16, then f possesses the angular limit c at eie. Consequently, a bounded holomorphic function can possess only one point asymptotic limit at eie. The first result shows that a bounded normal light interior function can possess point asymptotic limits at almost every point of C and possess no radial limits. The second result shows that a bounded normal light interior function can possess uncountably many distinct point asymptotic limits at the point z = l. THEOREM 9. There exists 3 bounded normal light interior function f 1p. D which possesses point asymptotic limits 35 plmost every point 2:. C but which possesses pp radial limits. Before proving the theorem we establish the following lemma. LEMMA 2. There exists 5 homeomorphism h f D. onto D with the following properties: (a) the radius To ‘35 z = l ip_mapped onto 2p are re, where F0 i§_g_Jordan arc lying 13 D U {1] internally tangent pp C at z l, with one end point 0, (b) .ifi Fe denotes the image pf F0 under §_rotation through 2p angle 6 about the origin, then the radius T9 ‘33 i e 6 ip_mapped onto F (c) the restriction pf h to C is the e, identipy and (d) h .35 HUC ‘ip D. 22 Proof. Let {Rn} be the sequence of real numbers constructed in Lemmaln Define the mapping 02 of the interval [0,R3) onto [0,R2) by §2(r) = (rR2)/R3. And define the mapping 9n (n = 3,4,...) of the interval [Rn,R onto [Rn-I’Rn) as in n+1) Lemma 1. Define the mapping Y of the interval [0,R3) onto 2 [0,1) by Y2(r) = p(0,r)/p(O,R3); and define the mapping Tn (n = 3,4,...) of the interval [Rn,Rn+1) onto [0,1) by Wu = p(Rnar)/D(Rn,Rn+1)- * Let C ={z: Im z>O, Iz -%\ =%] and let c- -| “R Lt c* 0- d1 - n {2. 2| n]. e n n wn an et an arg(wn) (n = 1,2,...) and a0 = 0. Define the mapping h in D by h(z) h(reie) = §n(r)eXP(iG + mm, + i(ozn_1 - an_2)‘i’n(r)) for O S r < R (n = 3,4,...); and 3 e16. It is easy to verify that h is homeomorphism of (n = 2), and RD s r < Rn+1 h(eie) D. onto D: By reasoning similar to that in Lemma 1 it is easy to verify that h is HUC in D. Setting To = h(TO) it follows that h possesses all the desired properties and the proof is complete. Egppf pf Theorem 9. Let h be the homeomorphism and To be the Jordan arc of Lemma 2. By a theorem of Lohwater and Piranian [16, Theorem 9, p. 15], there exists a bounded holo- morphic function g in D which does not approach a limit as 2 approaches eie along Fe (O s 9 < 2n). Hence the bounded light interior function f = g o h possesses no radial limits. By Theorem 6, f is normal. 23 Since g is bounded, g possesses radial limits at almost every point of C. Let T9 be the radius terminating at eie. It follows easily that f has point asymptotic limits at almost every point of C along the paths h-1(¢e); and the proof is complete. THEOREM 10. There exists 5 bounded normal light interior function f 1p. D which possesses chontinuum pf distinct point asymptotic limits pp the point z = l. Proof. Define the mapping f in D by f(2) = f(x + iy) = x + iy/(l - 23);:- It is easy to verify that f is a homeomorphism of D onto the unit square Q = {z = x + iy: max{lxl,|y|] < 1]. Let 2, z' E D, z = x + iy, z' = x' + iy' with z # z'. 35 we obtain ‘f(z') - f(§)] = lx' + iy'KLl - x'z)25 2 From the fact that Iy‘ < (l - x ) - x - iy/(l - x2)%] x + iy' - x - iy z - z x + iy' - x - iy 1 . y' - y I S 1 + (1 _ x'2)% ‘x' +-iy' - x - iy I + M ,‘ t1 - x2)” - <1 - x'zifl [(1 - x'2)<1 - x2)? 1 < 1 + 2 % (1-X') l 2 + 21; 2% (1-X').(1-X') E + (l - x2) And it follows that 24 M(f(z)) = lim sup Isz;% : :(Z)I < 2 . z' a z 1 - ‘zl And we obtain p*(f(z)) = M(f(z)) 2 --;;--§ . 1+|f(z)| 1- |z| of Lappan [14, Theorem 3, p. 156], f is normal. By a theorem For a (-l < a < 1) fixed, the Jordan arc 2 z(t) = t + ia(l - t )8 (O s t < l) in D is a point asymptotic path at z = l for the value 1 +-ia. Therefore f has a con- tinuum of distinct point asymptotic limits at the point z = l and the proof is complete. 3. Riesz's theorem Riesz's theorem [9] states that if f is a non-constant bounded holomorphic function in D and c is a fixed value in Q, then the set of points on C for which cA(£,e) = {a} cannot have positive measure on C. The following theorem shows that there exists a bounded normal light interior function f in D with CA(f,e) = {1] for every 9 6 [0,2n). THEOREM 11. There exists 3 bounded normal light interior function f ip D such that CA(f,9) = {1] for every 9 6 [0,2TT)' Proof. Let h be the homeomorphism of D onto the unit square Q which was constructed in Theorem 10, i.e. h(z) = h(x + iy) = x + iy/(l - x2)%. Define the mapping H in Q by H(z) = H(x + iy) = x(l - yz)!5 + iy. 25 It is easy to verify that H is a homeomorphism of Q onto D, and that H is uniformly continuous with respect to the chordal metric in both Q and .D. Since h is normal it follows easily that the homeomorphism H(h(z)) = x[(1 - x2 - y2)/(l - x2)]25 + iy/(l - x2)35 is a normal homeomorphism of D onto D. It is easy to verify that C(H o h, e) = {i} for all e E (0,n) and that C(H o h, e) = {-i] for all e E (n,2n). For a (-m < a < a) fixed, the segment z(t) = t + ia(l - t) in D terminating at the point z = l is mapped onto H(h(2(t))) = t[1 - 02(1 - t)/(1 + t)]25 + ia[(1 - t)/(1 + t)]%, so that H(h(z(t))) a l as t a 1. Hence, it is easy to verify that CA(H 0 h, 0) = [1} and similarly that CA(H o h, n) = {-1}. Therefore CA(H o h, 9) 6 {1, -l, i, -i] for 9 E [0,2n). Let g(z) = zh, then g is continuous on D U C and by Theorem 7 the light interior function f = g o h is normal in D. But g was choosen So that CA(f,e) = {1] for every 6 E [0,2n) and the theorem is proved. The previous theorem shows that the total outer angular cluster set of a light interior function can be a single point. If we consider the total cluster set instead of the total outer angular cluster set we are able to establish the following result. 26 THEOREM 12. Let f b ‘5 light interior function ip D which omits the value a. _f_ C(f) = {c}, 5322' c ==m. .EEEEE- Suppose C(f) = [c] and c¥=m then we have two cases: ‘Qggg i. f = g o h where h is a homeomorphism of D onto D, and g is a non-constant holomorphic function in D. Then C(g) = {c}, hence by a theorem of Lusin and Privalof [19, p. 72], g a c which is a contradiction. Eggs ii. f = g o h where h is a homeomorphism of D onto 0, and g is a non-constant holomorphic function in 0. Now g holomorphic in Q and g(z) a c #=a as 2 d a implies g is bounded in 0. By Liouville's theorem, g E c which is a contradiction. Therefore we cannot have C(f) = {c} for c # m and the theorem is established. A homeomorphism h of D onto 0 is a normal light interior function in D which omits the value a and C(f) = {a}. Therefore the hypothesis c # a was necessary for the previous theorem. The previous theorem is sharp as shown by the following result. THEOREM 13. There exists 2 bounded normal light interior function f ip D such that C(f,e) = {0] for every 9 E (0,2n). Proof. Let h be the homeomorphism of Theorem 11, i.e. h(z) = h(x + iy) = x[(1 - x2 - y2)/(l - x2)]% + iy/(l - x2)%. Then h is a normal homeomorphism of D onto D for which 27 C(h,e) c[n/2,311/2] for e e (0,2n). There exists a conformal mapping S1 of D onto the square Q = {2: O < x < 2, -1 < y < 1}, such that the arc [n/2,3n/2] of C is mapped onto the side L = {z E Q} x = 0] of Q: The mapping 82(2) = x + ixy/2 is a homeomorphism of Q onto the triangle T = [2: O < x < 2, IyI < x/2] such that the side L of Q. is collapsed to the point 0. The function 82 0 S1 is continuous on C U D, and by Theorem 7 the bounded light interior function f = S2 0 S1 0 h is normal in D. By construction we have C(f,9) = {O} for every 9 E (0,2n) and the proof is complete. IV. GENERALIZATIONS AND APPLICATIONS TO K-PM FUNCTIONS l. Quasiconformal functions and pseudo-meromorphic functions We now investigate the behavior of a light interior function f with StoIlow factorization f = g o h when h is a quasiconformal homeomorphism of D onto D and g is a non-constant meromorphic function in D. Let Q be a simply connected region in 0 bounded by a Jordan curve, and let 21, 22, 23, 24 be four distinct boundary points of Q, which lie in this order on the positively oriented boundary curve. We call such a configuration a quadrilateral, and denote it by Q(zl,22,23,24). An orientation preserving homeomorphism of the plane transforms quadrilaterals into quadrilaterals. Map the region Q conformally onto a rectangle R: 0 < u < l, O < V‘< t, in the w = u + iv plane, in such a manner that 21, 22, 23, 24 correspond to the vertices w = O, l, l + it, it respectively. We call the positive number t the modulus pf. the quadrilateral Q(z and denote it by 1,22,23,24), mon(z1,zz,23,z4). A homeomorphism h of D onto D is called K-guasi- conformal or simply KrQC, if i) h preserves orientation of the plane, and ii) for any quadrilateral Q(zl’22’23’z4) contained in D together with its boundary, mod h(Q(zl,zz,z3,z4)) S K mon(zl,zz,Z3,24)a 28 29 where K is a constant, K 2 1. If f is a light interior function in D with StoIlow factorization f = g o h with h a KEQC homeomorphism of D onto D and g a non-constant meromorphic function in D, then we shall call f a K-pseudo- meromorphic function, or simply K-PM. 2. Normality If f is a K-pseudo-meromorphic function in D with Stoilow factorization f = g o h, then we show that f is normal in D if and only if g is normal in D. This result was proved by Vgisala [21, Theorem 5, p. 20] whose proof is considerably different. THEOREM 14. f h is‘g KrQC homeomopphism pf D onto -1 D, then both h and h are HUC. e THEOREM 15. Let f .g‘Kspseudo-meromorphic function II 12. D with Stoilow factorization f = g o h where h is'p KrQC homeomorphism of D onto D and g i g non-constant meromorphic function ip_ D. .ngp_ f ‘i§_normal i2. D 'if‘gpg 29.11.!- g _'_s_normal_i_p D. _§£gp£ pf Theorem 14. Since h is KéQC, by a theorem of Mori [17] h.1 is also 'KEQC. Hersch and Pfluger [11] have shown that if h is KEQC then p(h(z),h(z')) s YK(p(z,z')) where YR is continuous and strictly increasing and defined for all x 2 O with YK‘O) = 0. It follows easily that h is HUC. Similarly h"1 is HUC and the theorem is proved. 2599;,gf_Theorem 15. From Theorem 14 both h and h“1 are HUC. By Theorem 6, f is normal in D if and only if g is normal in D and the theorem is proved. 3O 3. Preservation of Stolz domains Let h be a homeomorphism of D onto D. If for every 6 C and every Stolz domain A at e19 the image of some eie terminal Stolz domain of A is contained in a Stolz domain, then we shall say that h weakly_preserves Stolz domains. If h weakly preserves Stolz domains then we note that h has radial limits everywhere, and that CA(h,9) = CT (h,e) for .every 9 E C, where Te is the radius at 619. 6The following result is a generalization of Lindelgf's theorem [9]. THEOREM 16. Let f ,pglg light interior function i3, D II with Stoilow factorization f = g o h where h i§'§_homeo— 1' morphism pf. D onto D for which both h and h.1 weakly preserve Stolz domains and g is‘g mop-constant normal mero- morppic function ip, D. ‘lf f has the point apymptotic limit i i 9 e 0. c pp. e , then f has the angular limit c [pp Before we prove the theorem we establish the following lemma. LEMMA 3. _I_f h igphomeomorphismpi D onto D for which both h and h"1 weakly preserve Stolz domains, then h can pp_extended t ‘3 h_me9morphism‘p£ D onto D. Proof. Suppose h cannot be extended to be continuous in DI Then there exists a point e19 6.6 such that C(h,e) = [a1,a2], with O < a2 - a1 s 2n. There exist two radii id i¢ $1 and T2 terminating at e 1 and e , respectively, with 3 ch that h’1( i¢1 19 d a1<¢1<¢2 0, h(Jn)c{z: 1 - e< I2] < 1] for all but finitely many n, and the end points of h(Jn) tend to en: and eiB. Choosing a subsequence of {h(Jn)} if necessary, we may assume that there exists a Koebe sequence of arcs {Ln} relative to either the open arc (0,6) or the open arc (B:a + 2n) such that Ln<: h(Jn). But h-1(Ln)<: Jn so that {h-1(Ln)} is not a Koebe sequence of arcs in D, which contradicts our hypothesis that h- preserves Koebe sequences of arcs. Therefore h can be extended to be continuous in D“ and similarly h.1 can be extended to be continuous in D: By considering h o h.1 and h.1 o h it is easy to see that h can be extended to a homeomorphism of D. onto D. and the lemma is proved. Proof pf the Corollary. Since f is a non-constant normal K-pseudo-meromorphic function in D, f has the StoIlow factorization f = g o h where h is a KrQC homeomorphism of D onto D and g is'a non-constant meromorphic function in D. By Theorem 15 g is a normal meromorphic function in D. By a theorem of Mori [17, Theorem 4, p. 67], if h is a KEQC homeomorphism of D onto D, then h can be extended to a 34 homeomorphism of D onto D: Thus by Lemma 4 both h and h-1 preserve Koebe sequences of arcs. From Theorem 17, f possesses no Koebe limits and the proof is complete. V. ASYMPI‘OTIC BEHAVIOR The asymptotic behavior of a light interior function f in D with StoIlow factorization f = g o h is closely related to the asymptotic behavior of its component factors 3 and h. DEFINITION 4. Let f 29.5 function 22, D. e define the set A(f) ‘gp follows: ei9 E A(f) i£_there exists pp asymptotic path f f .23 D with end E and e19 6 E. Wg_also define the set Ap(f) .51 follows: e19 E Ap(f) .li there exists 5 point i6 asymptotic path pf f .lfl D terminating pp e DEFINITION 5. Let h ppmg homeomorphism pf D onto D. Wp_define the set B(h) ‘gp follows: e19 6 B(h) .i£ there exists pp grp,a§ymptotic path p£_ h .22. D ‘yipp.gpd E 33g e19 E int E, yhgrg! int E lg the interior pf E. THEOREM 18. ‘Lpp f 'pg 3 light interior function ip‘ D High StoIlow fpgtorization f = g o h ypgrg h is p_homeomorphism ‘2: D onto D and g i_flg non-constant meromorphic function pp D with A(g) dense pp C. Then A(f) U B(h) ip dense ip C. Proof. Suppose A(f) is not dense on C. Let (¢l,¢2)<: C - A(f) be arbitrary and let ,[91,92]<: (¢1,¢2), with O < 92 - 614< 2n. Let F1 and F2 be Jordan arcs in D at i9 i9 e 1 and e , respectively, with [F1 n F2 = [0}. Consider the domain A bounded by FI’U P2 and the arc [91,92] of C. 35 36 Then h maps A onto a domain R in D. Case i. CF (h,61) 0 CF (h,92) # ¢ and 1 2 [in a] = 01.101.91) U Cr2(h’92)' Let end 6 CF (h,el) n CF (h,92). There exist sequences 1 2 {2n} and {2;} in F1 and F2, respectively, Such that h(zn) ‘ fit“ and h(zé) * eia. Let A be a Jordan arc at eia which passes consecutively through the points h(zl),h(zi),h(zz),h(zé),... . By a lemma of Collingwood and Cartwright [7, Lemma 1, p. 93], either [01,92]<: CA(h-1,a) or [62,91 + 211] C CA(h-1,or). Hence, either (01,92) C B(h) or (92.91 + 2n) :2 B(h). Case ii. (h,6 ) n (h,e ) # ¢ and —— Crl 1 Crz 2 [in c] 01.9 ) u c (M >. i Crl 1 1‘2 2 Then E = [RIO C] - [CF (h,el) U CF (h,92)] is a non-empty . 1 2 open subarc of C. Let ela E E with ehy E A(g). Then ela is in the end of an asymptotic path A of g. But C(h-1,a)<: [91,92], and hence h-1(A) is an asymptotic path of f whose end intersects [01,62]. Thus [91,02] n A(f) # ¢ in violation of our assumption. Lass, iii. cr1(h’°1) n 3301,92) = d. Then E = [R:n C] - [CP1(h,91) U Cr2(h,02)] is a non-empty open subarc of C and by arguing as in Case ii we arrive at a contradiction. 37 Therefore, from.the above considerations, if c - , . ' , (¢1,¢2) C A(f), then (¢1 ¢2) n B(h) # ¢ Since (¢1 ¢2) was arbitrary it follows that A(f) U B(h) is dense on C and the proof is complete. THEOREM l9. .Lg; f ,hg.g.ligh; interior function ip D II with Stoilow factorization f = g o h where h 'ip p homeo- morphism pf, D onto D with Ap(h) dense pp D, and g is a non-constant meromorphic function 1p. D with Ap(g) dense pp C. Then Ap(f) U B(h) 1p dense pp C. Proof. Suppose B(h) is not dense on C. Let (¢1,¢2) C C - B(h) be arbitrary and let [61,92] C (¢1,¢2) 161 i6 with °<°2"°1<2” and e ,e 2€Ap(h). Let I" and l 101 F2 be two asymptotic paths of h terminating at e and i9 e 2, respectively, with F1 n F2 = [0}. Consider the domain A bounded by F1 U P2 and the arc [61,92] of C. Then h maps A onto a domain R in D. We have [Rim C] # [ely], since [61,62] fl B(h) = ¢. Hence, E = [Rim C] - [Cr1(h’91) U Cr2(h,62)] is a non-empty open subarc of C. Since Ap(g) is dense on C there exists a point ehy E E and an asymptotic path A of g terminating at eh”. But C(h-1,a)'C:[31,62], hence h-1(A) is an asymptotic path of f whose end intersects [91,92]. If h- (A) ends in a subarc [31,32] of [91,62], then our assumption that [01.92] h B(h) = ¢ is violated. Thus, h-1(A) ends at a point e19 6 [91,92] n Ap(f). Therefore, if (¢I,¢2)<: C - B(h), then (¢1,¢2) n Ap(f) # ¢. Since (¢1,¢2) was arbitrary we have that Ap(f) U B(h) is 38 dense on C and the proof is complete. COROLLARY. Let f .23.2 light interior function 1p D with StoYlowfpctorization f g o h where h lp'p homeomorphism ‘pg D onto D for which both h and h.1 are HUC, and g .pp .p mpg-constant meromorphic function ip D .EEEH Ap(g) ‘ppppp pp C. ,3233 Ap(f) U B(h) .EE.EEE§S.EE C. The corollary will follow immediately from the previous theorem when we establish that Ap(h) is dense on C. The proof of the corollary will be complete when we establish Lemma 6. LEMMA 5. Let h ‘pp p homeomorphism pf D onto D which is HUC. Thpn, for any 6 > 0 there corresponds p constant K > 0 such that p(h(z)),h(z')) < K.p(z,z') for every pair pf points. z, z' E D with p(z,z') 2 6. Proof. Suppose that the lemma is false. Then there exist d ' ' d two sequences {2“} an {2“} such that p(zn,zn) 2 6 an p(h(z ),h(z')) 2 n p(z ,z'). Let l/ZN < 6 and let P be the n n n n n non-Euclidean geodesic joining zn to z; (n = N, N+l,...). Partition F by points z , (j = l,2,...,m ) such that n n,j n I z = z and z = z and l/2n < p(zn z n n,l n n,mn ,j’ n,j+l (j = l,2,...,m -l). There exists an integer k (l s k .< m ) n n n n ) < l/n . h such that p(h(zn,kh)’h(zn,kn+1)) 2 n p(zn ) T en ,k ’zn,k +1 n n and {z are close sequences with {zn,k } n,k +1} n n p(h(zn 9 k )3h(zn,kn+1)) 2 n p(zn,kn’zn,kn+1) > 1/2 in Violation of our hypothesis that h is HUC. Therefore, the lemma is true. LEMMA 6. Let h be‘a homeomorphism pf D onto D for which both h and h”1 are HUC. Then both h and h"1 have radial limits everywhere. 39 Proof. It will suffice to show that b has radial limits everywhere. Without loss of generality we may assume that h(O) = 0, for otherwise we may consider the homeomorphism h (0) - h(z) l - h(O) h(z) H(Z) = where both H and H"1 are HUC. From Lemma 5 we can find a smallest integer K for which p(h(z),h(z')) < K.p(z,z') and p(h-1(z),h-1(z')) < K p(z,z') whenever p(z,z') 2 1. Construct a sequence {Rn} of real numbers, 0 = R < R. < R. <...< R.n <...< l, 0 l 2 such that p(O, Rn) = n. Let An = {2: RH s [z] s Rn+l}' Then H(A.) CZ U Aj’ for some integer N. Since h(O) = O we have j=O l/K p(O,z) < p(0,h(2))‘< K p(0,z) for all z E An (n = N+1,N+2,...). Notation convention. The subscript Kn in RKn is the integer j = Kn, and is not to be confused with double subscripts. Let B (0 s B < 2n) be fixed but arbitrary. Set M = max[N+l, K2 + 2]. Then p(O,RKn) = Kn = UK p(O,R 2 e18) < p(O,h(R 2 e18» K n K n (n = M, M+l,...), and we obtain the inequality (1) RKn< |h(RK2neIB)‘ and RKn< Rxm +1) <|h(1iK2 (MI) 616)‘ (n =‘M, M+1,...). 18 Set an = arg h(R 2 e ) (n = M, M+l,...). Then K n 4O p(RKnemnaK eflan+1) s p(h(R 2 eiB>.h> “ K n K (n+1) I I < K.p(R 2 ,R 2 ), and K n K (n+1) I id 3 n m’n-i-l ~ a) pan; ,gme )<:K 2 = K p(R .R 3) Kn - K - K Kn - K (n = M, M+l,...). id Let Fn be the non-Euclidean geodesic joining RKne n fly to RKne n+1. Then from (1) and (2) it is easy to see that (3) min [‘2]: 2 E F ]‘2 R 3. “ Kn - K Using (3) we obtain the inequality by by n n+1 _ d2 <4> wee > - I 4—1—2 F 1 - ‘2‘ n R 3 M l I > O’ " 0’ 1 _ R2 3 n+1 n Kn-K Using (4) and (2) it follows that . fly fly, 2 n n+1 Ia - a I s (<1 - R >/R )p( e , e ) n+1 n Kn-K; Kn-K3 RKn RKn 2 2 Kn-K Kn-K -K Kn-K Kn-K s K2(R 3 - R )/R 3 Kn-K K(n-1)-K Kn-K3 41 Therefore we obtain the inequality m «:2 (5) 2‘6! -aI l there-exists a K-quasi-conformal homeomorphism h of D. onto D, such that a certain set E of linear measure zero on C is mapped onto a set F of linear measure Zn on C. 42 By a theorem of Lohwater and Piranian [16, Theorem 4, p. 11], there exists a bounded holomorphic function g that has no radial limits at points of the set S = C - F. By a theorem of Lehto and Virtanen [15, Theorem 2, p. 53], g has no point asymptotic limits at points of S. By Theorem 15, f = g o h is a normal light interior function. But Ap(f)<: E, hence 0. 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