NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS
NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS
CHI-TAl CHUANG Department of Mathematics Peking University, China
lh World Scientific '
. . . , Singapore· New Jersey· London· Hong Kong
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NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS Copyright © 1993 by World Scientlfic Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any fonn or by any means, electronic or mechanical, including photocopying, recording orany infonnation storage and retrieval system now known or to be invented, without written permission from the Publisher.
ISBN 981-02-1257-7
Printed in Singapore by Utopia Press.
PREFACE The theory of normal families of holomorphic functions or more generally of meromorphic functions was created by Montel about eighty years ago. His definition of normal families of holomorphic functions may be stated as follows: Definition 1. A family of holomorphic functions in a domain D is said to be normal in D, if from every sequence of functions fn(z) (n=l,Z,···) of the family, we can extract a subsequence fn, (z) (k = 1, Z,· .. ) which converges locally uniformly in D to a holomorphic function or the constant
00.
In order to extend this definition to the case of families of merom orphic functions, it is necessary to give a definition of the locally uniform convergence of a sequence of merom orphic functions in a domain. Landau and Caratheodory first gave a such definition. Later Ostrowski pointed out that it can be equivalently defined by means of the spherical distance WI' W2
of the extended complex plane
I WI' W2 I
C=C U (00).
between two points
A locally uniformly conver-
gent sequence of meromorphic functions in a domain with respect to the spherical distance, has a limit function which is a meromorphic function in the domain or the constant
00.
So the notion of normal families of meromorphic functions can
be simply defined as follows: Definition Z. A family of merom orphic functions in a domain D is said to be normal in D, if from every sequence of functions fn (z) (n = 1, Z, .. ·) of the family we can extract a subsequence fn, (z) (k = 1, Z, .. ·) which is locally uniformly convergent in D with respect to the spherical distance. In particular for a family of hoi om orphic functions in a domain, this definition is equivalent to Definition 1. The theorey of normal families of meromorphic functions has various applications such as uniform convergence of sequences of hoi om orphic functions or meromorphic functions, extensions of Picard's theorems, Schottky's theorem and Landau's theorem, iteration of rational functions or entire functions and conformal mapping. Next to the notion of normal families of meromorphic functions, Montel al-
Preface
vi
so introduced the notion of quasi-normal families of meromorphic functions which may be defined as follows: Definition 3. A family of merom orphic functions in a domain D is said to be quasi-normal in D, if from every sequence of functions fn (z) (n = 1, 2, ••. ) of the family, we can extract a subsequence fn, (z) (k = 1,2, ••• ) which is locally uniformly convergent in D-cr with respect to the spherical distance, where cr is a set of points of D having no point of accumulation in D and depending on the sequence of functions fn(z) (n=1,2,···). Chuang observed that if we denote by E the set of points of D in the neighborhood of each of which the extracted sequence fn, (z) (k = 1,2, ••• ) is not uniformly convergent with respect to the spherical distance, then normality and quasi-normality of a family of merom orphic functions in D correspond respectively to the conditions E =
0
and E' =
0,
where E' is the derived set in D of E. It
is natural to go still further and introduce notions corresponding respectively to the conditions E"= 0 ,E"'= 0
, ....
In doing so, for each integer m;?:O, Chuang
defined a corresponding notion of Qm-normality corresponding to the condition E (m) =
0,
such that in particular for m = 0 and m = 1, it coincides respectively
with the notions of normality and quasi-normality defined respectively by Definitions 2 and 3. The theory of quasi-normal families of meromorphic functions and more generally that of Qm-normal families of merom orphic functions also have various applications. This book consists of nine chapters and two appendices. They as a whole constitute a systematic general theory of normal families, quasi-normal families and Qm-normal families of meromorphic functions with many applications. Many materials in this book are research works of the author unpublished before. Chapters 1, 2 and 3 are concerned with normal families of merom orphic • functions. They contain mainly a precise definition and three necessary and sufficient conditions of the normality of a family of merom orphic functions in a domain, some standard criterions of normality and their applications. Chapter 4 deals with closed families of merom orphic functions. A closed family of meromorphic functions in a domain is a normal family in that domain, but the converse is untrue. By means of continuous functionals and continuous operators defined on a closed family of merom orphic functions, for many extremal problems, the existence of the solution can be deduced. A covering theo-
Preface
vii
rem concerning the existence of an extremal domain covered at least p times by the values of each function of a closed family of meromorphic functions satisfying a certain condition, is proved. Chapter 5 is concerned with quasi-normal families of merom orphic functions. Besides a precise definition and some properties of such a family, this chapter consists mainly of two parts. The first part deals with quasi-normal families of order not exceeding an integer v. Correspondingly, criterions and a necessary and sufficient condition of quasi-normality of order not exceeding an integer v
are obtained, with applications. The second part deals with quasi-normal fami-
lies of finite total order. After some preliminary preperations, the notion of such families is defined. This is then followed by some examples and applications. The main purpose of Chapters 6 and 7 is to give applications of the theory of normal families or quasi-normal families of meromorphic functions, which are not contained in the preceding chapters. However the criterions of normality or quasi-normality of families of holomorphic or meromorphic functions proved in Chapter 7 are also of importance. Chapter 8 is concerned with the theory of Qm-normal families of merom orphic functions. It consists mainly of the definition and necessary and sufficient conditions of Qm-normality, the definition and necessary and sufficient conditions of Qm-normality of order not exceeding an integer v, criterions of Qm-normality or Qm-normality of order not exceeding an integer v and an existence theorem which justifies the introduction of the notion of Qm-normal families of merom orphic functions. The theory of Qm-normal families of meromorphic functions has various applications which are given in Chapter 9. In Appendices A and B are proved some theorems which are used in the proofs of some criterions of normality or quasi-normality in Chapters 2,3 and 5. The author has tried to write this book as simply and as clearly as possible. For instance, in this book the definitions of normality and quasi-normality of a family of meromorphic functions are in substance equivalent to those given by Montel, but in a different form which is much clearer and easier to handle. However the proofs of some theorems are long. In order to facilitate the reading, they are given in detail. The author also has tried to keep this book at an elementary level. It is self-contained, except only that in Appendix B some elementary knowledge of Nevanlinna' s theory of meromorphic functions is needed, which can be found from anyone of the following three books:
Preface
viii
Nevanlinna. R .• Le Theoreme de Picard-Borel et la Theorie des Fonctions Meromorphes. Gauthier-Villars. Paris. 1929. Hayman. W. K .• Meromorphic Functions. Clarendon Press. Oxford. 1964. Chuang. C. T. and Yang. C. C •• Fix-Points and Factorization of Meromorphic Functions. World Scientific Publishing Co .• Singapore. 1990. It is assumed that the reader is familiar with the basic theory of functions of a
complex variable and has some elementary knowledge of Nevanlinna' s theory of meromorphic functions. For reference. the reader may consult the following books: Montel. P .• Lecons sur les familles normales de fonctions analytiques et leurs applications. Gauthier-Villars. Paris. 1927. Valiron.
G..
Familles
N ormales
et
Quasi-normales
de
Fonctions
Meromorphes. Memorial des Sc. Math .• Fasc. 38. 1929. Gu. Y. X .• Normal Families of Meromorphic Functions (in Chinese). Educational Press of Szechuan. 1991. Unlike the first two books. the third book deals only with normal families of meromorphic functions and the emphasis is placed on generalizations of a theorem of Miranda. The References at the end of this book comprise only those papers and books quoted in the text. with few exceptions. For other references the reader may consult the above three books. Finally I express my sincere thanks to Mrs. Hu Yang and Mrs. Wang Hui Li for their efforts to arrange to transform my manuscript into a camera-ready manuscript by the Computer Centre of Peking University. I thank especially Ms. Yan Ling and Mr. Yan An of that centre who undertake this laborious
task. I am also very grateful to Dr. K. K. Phua, Mr. J. G. Xu and the staff of the World Scientific Publishing Company for the publication of this book.
Chi-tai Chuang
CONTENTS
Preface Chapter 1. Basic Notions and Theorems
1
1. 1. Spherical distance
1
1. 2. Sequences of merom orphic functions
4
1. 3. Families of meromorphic functions
13
Chapter 2. Criterions of Normality of Families of Holomorphic
27
Functions and Applications 2. 1. Montel's theorem
27
2.2. Miranda's theorem
40
2. 3. Bloch's theorem
51
Chapter 3. Criterions of Normality of Families of Meromorphic
57
Functions and Applications 3.1. Montel's theorem
57
3. 2. Zalcman's theorem
73
3. 3. Gu's theorem
86
Chapter 4. Closed Families of Meromorphic Functions
93
4. 1. Closed families of holomorphic functions
93
4. 2. Examples
101
4.3. Closed families of meromorphic functions
106
x
Contents
4. 4. Examples
118
4.5. A covering theorem
123
Chapter 5. Quasi-normal Families of Meromorphic Functions
131
5. 1. Some preliminary definitions and theorems
131
5.2. Criterions of quasi-normality of families of meromorphic functions
136
5. 3. Some applications of criterions of quasi-normality
166
5. 4. Quasi-normal families of meromorphic functions of finite total order
183
5. 5. Applications
203
Chapter 6. Further Applications
219
6. 1. Uniform convergence of sequences of meromorphic functions
219
6.2. Generalizations of Bloch's theorem and some theorems of Valiron
238
6.3. Univalent and multivalent functions
249
Chapter 7. Extensions of Some Criterions of Normality
273
and Quasi-normality 7.1. Case of holomorphic functions
273
7.2. Case of meromorphic functions
281
7.3. Case of univalent and multivalent functions
303
Chapter 8. Qm-normal Families of Meromorphic Functions
309
8. 1. Some notions and theorems
309
8. 2. An existence theorem
314
8. 3. Necessary and sufficient conditions for Qm-normality
318
8. 4. Conditions and criterions of Qm-normality
330
8.5. Other criterions of Qm-normality
341
Chapter 9. Applications of the Theory of Qm-normal Families
347
of Meromorphic Functions 9. 1. Uniform convergence of sequences of meromorphic functions
347
Contents 9. Z. Distribution of the values of merom orphic functions in the
xi
354
neighborhood of an essential singularity 9.
3. Limitation of the modulus of meromorphic functions
359
9. 4. Limitation of the number of the zeros of meromorphic functions
379
9. 5. Domains covered by the values of meromorphic functions
383
Appendix A. A General Theorem on Holomorphic Functions in
393
the U nit Circle
1. A theorem on convex functions
393
Z. A general theorem on holomorphic functions in the unit circle
401
3. Generalization of theorem Z
435
Appendix B. Some Theorems on Meromorphic Functions
443
1. A theorem on meromorphic functions in the plane
443
Z. A theorem on merom orphic functions in the unit circle
445
References
465
Index
469
1 BASIC NOTIONS AND THEOREMS
1. 1. SPHERICAL DISTANCE In the figure, the equation off the sphere S is
x
2
+ + (u y2
-
1
2-)2
=
1
4'
(1. 1)
Consider a complex number z=x+iy. Let p be the point of the xy plane corresponding to z,
whose coordinates are
(x, y).
The
straight line joining the two points Nand p intersects S at a point m distinct from N. We call m the point of S corresponding to z. Let us find out the coordinates (X, Y ,u)of m. We have
x
= hx,
Y = hy,
u -
1 = - h,
where h is a positive number. Substituting into 0.1) we get 1
and
x X=1+lzI 2 '
Y y -1+lzI 2 '
_ U -
1
Iz 12 Iz
+
12 '
(1. 2
Normal Families of
2
Meromorph~c
Functwns
The point of S corresponding to 00 is the point N whose coordinates are (0, 0,1).
Definition 1. 1. Let Zl ,Z2 be two points of the extended complex plane C=C
U 00
and ml' m2 the two points of S corresponding respectively to Zl' Z2. The
length of the line segment mlm2 is defined to be the spherical distance between ZpZ2 and is denoted by IZPZ21. Let us find out an expression of IZl ,z21. Distinguish three cases: l)ZpZ2 are both finite. Let zj=xj+iYj(j= 1, 2) and set k j= 1 + IZj 12(j= 1,2). By
o. 2),
we have
(klk2IzpZ21)2 = (k2Xl -
=
k1X2)2 + (k 2Yl-k 1Y2)2 + (k 1 -
k22kl + k/k 2 -
k2)2
2k 1k 2(XIX2 + YIY2 + 1)
and hence (1. 3)
Next using the relations
we find that the right member of (1. 3) is equal to IZl -z21 2. So we have the formula
0.4)
2)One of Zl ,Z2 is finite and the other is infinite. For instance Zl =Xl +iYl is finite and Z2=00. Then
IZH Z21 2 = 0 +
xi IZ112)2 + (1 + 1
Yr 1 IZ112)2 + (1 + IZ112)2
3
Ba81,c Notions and Theorems
and hence
1
0.5)
3)zl ,Z2 are both infinite. Evidently IZI ,z21 =0. By Definition 1. 1 the triangular inequality
0.6) obviously holds for any three points Zj (j = 1,2,3) of
C. Also
it is easily verified
that the formula
(1. 7)
holds for any two points zj(j=1,2) of
C.
Finally we prove two lemmas which are sometimes useful.
Lemma 1. 1. Let ZI ,Z2 and a::;i:oo be three points of C. Then (1. 8)
Proof. Assume first that Zj::;i:oo(j=1,2). In this case, 0.8) follows from the formula
and the inequality 1
+
lSI - s21 2 = 1
1. Then we can apply the result obtained in the first case to the
27
28
Normal Families of Meromorphic Functions
function 1/f(z), hence in the circle
f, we have 11/f(z) I<e h •
Consequently by Corollary 1. 4, in the particular case a=O, b=l, the family57is normal in D.Now consider the general case. Let fn(z)(n=1,2,"')be a sequence of functions of 57. Define ( ) _ f.(z)-a b-a
(n = 1,2",,),
g. z
Since gn (z) does not take the values 0 and 1, from the sequence gn (z)(n = 1,2, ···),we can extract a subsequence gn,.(z)(k=1,2,···) such that as k-1+=, gn,. (z) converges locally uniformly to a holomorphic function or to = in D. Evidently this is also true for the subsequence
f •• (z) =
a
+ (b-a)g •• (z)
(k
= 1,2",,).
We are going to give some applications of Theorem 2. 1. For this, we need the following lemma:
Lemma 2. 1. Let 57 be a normal family of holomorphic functions in a domain D. Let a be a bounded closed set of points belonging to D and M a positive number. Assume that for each function f(z)E57, we have min If(z) ·Ea
I ~ M.
(2.1)
Then the family 57 is uniformly bounded on each bounded closed set E of points belonging to D.
Proof. Let E be a bounded closed set of points belonging to D. Assume that 57 is not uniformly bounded on E. Then to each positive integer n, corresponds a function f n(z)E57 such that max If.(z) .EB
I>
n.
(2.2)
Consider a subsequence fn,. (z) (k = 1,2, ••• ) of the sequence fn (z) (n = 1,2, ••• ). By (2.2) and (2.1), as k-1+=, fn,. (z) can not converge locally uniformly to a
Cnterions of N ormalitg of
holomorphic function or to
00
Fam~lies
of H olomorphic Functions
29
in D. This contradicts the hypothesis that 7
is a
normal family. In the applications of Lemma 2. 1, the set
(J
often consists of a single point.
Theorem 2. 2. Let f(z) be a non-constant entire function. Then the family of entire functions f.(z) = f(2'z) (n = 1,2,···)
(2.3)
in not normal in the circle Iz I0 de-
pending only on v having the following property: If f(z) is a holomorphic function in the circle Iz I 1 a number.
Let Y be the family of the functions f(z) satisfying the following conditions: 1 0 f(z) is holomorphic and does not take the value zero in D. 2 0 There is a number w=w(f) such that 0~w 0, (1
+
la/f) 12)'Z
laj(f)
1O,R. 21
1
1 + M2If,(z) + a. ,f,(zo) + a,l,
hence the family 571 is also equicontinuous in D with respect to the spherical dis-
92
Normal Families of
tance, and therefore normal in D.
4 CLOSED FAMILIES OF MEROM ORPHIC FUNCTIONS 4. 1. CLOSED FAMILIES OF HOLOMORPHIC FUNCTIONS Definition 4. 1. Let.'iT be a family of hoi om orphic functions in a domain D. We say that .'iT is closed in D, if from every sequence of functions f.(z)(n= 1, 2,,,,) of .'iT, we can extract a subsequence f.. (z)(k = 1 ,2, ... ) such that as k-1
+00, f.. (z) converges locally uniformly in D to a function Hz) E.'iT. Evidently a closed family .'iT is a normal family.
Definition 4. 2. Let.'iT be a closed family of holomorphic functions in a domain D. Let .(z)-lf'(z) 1= 1tJ>{g.(z)}-tJ>{G(z)} 1< e. Hence as n---1+ oo , tV. converges uniformly to \If in c'. Since the point Zo of D is arbitrary, as n---1+ oo , tV. converges locally uniformly in D to \If. Consequently the operator {Q (f)} is continuous. In particular,5) holds when (w) is an entire function, for instance (w) =e
W
•
Closed Families of Meromorphu: Functions
101
4. 2. EXAMPLES 1)Schottky' s theorem and Landau's theorem
Lemma 4. 1. Let wo#O, 1 be a complex number. Let
sr be the family of
holomorphic functions f(z) in the circle ~: Iz I 0 is said to have the property (P) for a funetion f(z) E 7 0, if there is a circle
IW-Wo IO is said to have the property (P) for 7 0• if b has the property (p) for each function f(z) E 7 0, Since the function zE70• evidently if a number b>O has the property (p) for 7 0, then b~1.
Theorem 4. 4. Let 70 be the family of holomorphic functions in
~ de-
fined in Definition 4.4. There exists a number B(Oto, the values of fo(z) do not cover at least p times the domain 6,. The proof consists of several steps: j) By Lemma 4.
7 and the condition (4. 13), we can find a circle IZ-Zo I toand assume, on the contrary, that the values of the function fo(z) cover at least p times the domain .:'.,. Then since woE:I"C .:'." the number of the zeros of the function fo(z)-wo in D is greater or equal to p. Hence we can find a finite n urn ber of dis tinct points Zj (j = 1, 2 , ••• , m) of D, such that Zj is a zero of order Aj of fo(z)-wo with m
~).j ~ p. j=1
Let Iz-zjl~rj(j=1,2,···,m) be m circles belonging to D, exterior to each other and containing no other zeros of fo(z)-woexcept zj(j=1,2,···,m). By Theorem 1. 1, we may assume that, for each j(l~j~m), the functions fn, (z)(k~kj) and fo(z) are holomorphic in the circle IZ-Zj I <rp and as k---1+ oo , fn, (z) converges uniformly to fo(z) in the circle Iz-zjlO.
~
Let K =K (mHm20 ••• ,
j==1
m. ,0) be the condition defined as follows: We say that a meromorphic function f (z) in a domain D satisfies the condition K in D, if f(z) satisfies the condition 2° in Theorem 3.13. By Theorem 3.13, this condition K is a condition of normality. Let e>O be a number. Consider a function f(z) of the family.'iT. By Theorem 5. 3, it is sufficient to show that f(z) has at most
• pj distinct (K ,e)-circles
~
j-1
in D. Assume on the contrary, that f(z) has p=
•
~pj+l
distinct (K,e)-circles
j-I
rl
(j=1,2,···,P) in D. Consider the q corresponding values aj(f)(j=I,2,···,q) satisfying the condition 2° in Corollary 5. 5. Denote by Uj the set of the roots of
•
the equation (5. 4), whose order of multiplicity is less than mj. Let E = Uaj. j-1
Then for each l~i~P, the circle
r l contains a point of E. Consequently E con-
sists of at least P distinct points, incompatible with the condition 2° in Corollary 5.5.
Corollary 5.6. Let D be a domain. Let ajEC(j=1,2,···,q;3~q~5) be qdistinct values. Let mj~2(j=1,2,···,q) and pj~O (j=I,2,···,q) have the same meaning as in Corollary 5. 5. Let .'iT be the family of the functions Hz) satisfying the following conditions: 1 ° f(z) is meromorphic in D. 2° For each l~j~q, the equation
has in D at most pj distinct roots whose order of multiplicity is less than mj.
142
Normal Fam,lies of Meromorph,c Functions Q
Then the family .']iT is quasi-normal of order L Pi at most in D. j=l
This corollary is a generalization of Theorem 3. 12.
Corollary 5.7. Let
D be a domain. Let M, b(OO. Let K = K (M ,b ,v) be the condition defined as follows: We say that a meromorphic function f(z) in a domain D satisfies the condition K in D, if f(z) satisfies the condition 2° in Theorem 3. 17. By this theorem, the condition K is a condition of normality. Let E>O be a number. Consider a function Hz) of the family.']iT. By Theorem 5. 3, it is sufficient to show that Hz) has at most p +q distinct (K, E)circles in D. Assume on the contrary, that Hz) has p+q+1 distinct (K,E)-circles r;(j=1,2,···,p+q+1) in D. Consider the two corresponding values a(f) and b (f) satisfying the condition 2° in Corollary 5.7. Then for each l~i~p +q +1, the circle r; contains a root of one of the equations (5.5), and hence the number of the distinct roots in D of the equations (5.5) is at least equal to p+q + 1, incompatible with the condition 2° in Corollary 5. 7.
Corollary 5. 8. Let D be a domain, a and b"eO two complex numbers,
QuaS/,-normal F amtltes of M eromorp hw Functwns
and v?l an integer. Let p?O and q?O be two integers. Let
143
sr be the family of
the functions f(z) meromorphic in D and such that the equations f(z)
=
a
and
f(v)(z)
=
b
have respectively at most p distinct roots and q distinct roots in D. Then the family
sr is
quasi-normal of order p+q at most in D.
Now we introduce the notion of irreducible C]-sequences, which plays an important role in the study of the order of a quasi-normal family of meromorphic functions.
Definition 5.6. Let S:fn(z)(n=1,2, .. ·) be a C]-sequence of merom orphic functions in a domain D and let Zo a nonCo-point of S in D. Then two cases are possible: lOWe can extract from the sequence S a subsequence S' of which Zo is a C opoint. In this case, we say that S is reducible with respect to the point Zo0 2 0 Zo is a nonCo-point of every subsequence of the sequence S. In this case, we say that S is irreducible with respect to the point Z00 The sequence S is said to be an irreducible sequence in D, if in D, S has nonC 0points with respect to each of which S is irreducible.
Lemma 5.5. Let S:fn(z)(n=1,2,"') be a C]-sequence of meromorphic functions in a domain D. Assume that S has nonCo-points in D. Then we can extract from S a subsequence S' which is either a Co-sequence in D or an irreducible C]-sequence in D. For the proof of this lemma, we need the following lemma of which we give a proof by a method already used in the proofs of Theorems 1. 4 and 5. 1.
Lemma 5. 6. Let domain D. If
(J
(J
be a set consisting of an infinite number of points in a
has no point of accumulation in D, then
(J
is enumerable.
Proof. Let Zj (j = 1 ,2, ... ) be a sequence of points of D, such that each point of D is a limiting point of this sequence zj(j=1,2,"')' Consider a point Zj and let R j be the least upper bound of the set of the positive numbers r such that
144
Normal Familtes of Meromorphtc Functions
the circle IZ-Zj I 1 for n~no.
Then by (5. 12) and
(5. 13), the sequence of functions
is locally uniformly bounded in D. Consequently from the sequence
%(z)(n~no)
we can extract a subsequence
g g'
Then from the identity 9 (Z)-w = {p (Z)-w}
+ p (Z)(l)(Z)
and the inequality
1
Ip(Z)(l)(Z)1 < 10' we conclude, by Rouche' s theorem, that the equation 9 (Z) = w
(5.26)
has only one root Z=ho(w) in Eo. The function Z=ho(w) is holomorphic and u-
Quam-normal Famtltes of Meromorphw Functtons
159
nivalent in H~ and satisfies the identity g (ho(w)} = w
(5.27)
in H~. If instead of the domain Eo, we start from the domain
R
8< Izl (Z)} ,
1 larg{l+tl>(Z)}1 <arcmn 10'
1
1f
0< "4-arcmn 10
1f
1f
< argg (Z) 1 for n~no. As in the proof of Theorem 5. 5, we see that the sequence of functions f.(z) cp.(z) = B(zo,f.)
(n ~no)
is locally uniformly bounded in D. Consequently from the sequence cp.(z) (n~no) we can extract a subsequence
f •• (z)
.
cp.. (z) = B (z 0 ,f.)
(h = 1, 2 , ••• )
such that as h---1+ oo , qJ"" (z) converges locally uniformly to a holomorphic function qJ(z) in D. As in the proof of Theorem 5.5, we see that qJ(z)~O and so we need only to show that qJ(z) has at most k-l distinct zeros in D. Assume, on the contrary, that qJ(z) has k distinct zeros z/j=1,2,···,k) in
D, of orders mj(j=1,2,"',k) respectively. Let Iz-zjl~rj(j=1,2,···,k) be k mutually disjoint circles belonging to D, such that for each l~j~k, we have
in the circle Iz-zj I~rj' Then we can find a number A>O and k circles Pj(O0 and 0>0 be two numbers. Assume that the following conditions are satisfied: 10 The family $T is quasi-normal of order vat most in D, where v~O is an integer. 20 For each function f(z) E $7 , there are v+ 1 points Zj =Zj (0 E a(j = 1,2, ... ,v+ 1) such that
(5.34) and that
If(z)1 ~M
(j = 1,2,···,v+ 1).
Then the family $T is normal in D. Consequently, by Lemma 2. 1, the family $T is uniformly bounded on each bounded closed set of points belonging to D. This lemma is a generalization of Lemma 2. 1. In this lemma it is tacitlyassumed that there exist v+ 1 points zjE a(j = 1,2, •.. ,v+ 1) satisfying the condition (5.34).
Proof. Let S dn (z)(n = 1,2, .•. ) be a sequence of functions of the family $T. By hypothesis, we can extract from S a subsequence S' : fn,. (z) (k = 1,2, ... )
of which the number of nonCo-points does not exceed v. Set
By Lemma 3. 2, we can find an increasing sequence of positive integers k p (p = 1 , 2, ..• ) such that for each
l~j~v+ 1,
we have
lim z(',) =
,-)+0::> )
where !;'jEa
s"J
(j=1,2,"',v+l) and
Isrs}'1
~o
O~j,j' ~v+l,j#-j').
Quam-normal Families of Meromorphtc Functtons
167
Consider the sequence S": fm, (z) (p = 1,2,·,,; mp = nt,) of which the number of nonCo-points also does not exceed v. We are going to show that SrI is a Co-sequence in D. This will complete the proof of Lemma 5. 10. In fact, assume that S"has nonCo-points ai(j=1,2'···'IlH.l~v) in D. In the domain DI =D-(al ,a2' ···a.) SrI is a Co-sequence. By a remark made at the beginning of this chapter, as p~+oo, fm, (z) converges locally uniformly to Among the points
~i (j =
1,2, ••• , v+ 1) there is at least one
~p
00
in DI"
for example, be-
longing to D I. T hen there is a circle IZ-~I IO such that l/K~~ and consider the function f 1(z)=(z"-I)/ n! K. Evidently f) (z) E71 and M (r ,ff'» = 1/K. Hence M (r ,F;n»;?1/K and we have necessarily F n(z) E 7).
Definition 5. 8. Let Hz) be a merom orphic function in a domain D. Let E be a bounded closed set of points belonging to D. Let q;?l be an integer and b >0 a number. We define, when q;?2,
a(E,q,o;j) = sup {mtn J(Zj,j)}, (f
(5.44)
l~j~q
where the sup is taken over all the systems a:
Zj
(j = 1,2,,,,, q) of q points such
that z j EE(j=1,2,"',q) and that
When q = 1, we define
a(E,q,o;f) =maxJ(z,f). zEE
(5.45)
176
Normal F amihes of Meromorp hic Functions
Lemma 5. 12. Let
.J}T
be a quasi-normal family of order v at most of
meromorphic functions in a domain D, where v? 0 is an integer. Let E be a bounded closed set of points belonging to D and let 1'»0 be a number. Then there is a positive number M such that for each function Hz) E.J}T, we have
a(E ,v
+ l,o;!) ~ M.
Proof. If v= 0, then the family
.J}T
(5. 46)
is normal in D, and the existence of the
number M follows from (5. 45) and Theorem 1. 6. Consider the case v?1. Assume that the number M does not exist. Then for each integer n?I, there is a function fn (z) E.J}T such that
a(E,v+l,o;f.»n and hence there is a system Zj(n) (j= 1,2, ... ,v+ 1) of points such that Zj(n) E E (j=
1,2, .. · ,v+1) ,that
and that
mln (](zj') ,f.) >n.
(5.47)
l~J~"+l
From the sequence fn (z) (n = 1,2, .. ·) we can extract a subsequence fn, (z) (k = I, 2, .. ·) which is a C1-sequence in D and has at most v nonCo-points in D. Consider the sequences of points zj"') (k = I, 2, ... ) (j = I, 2, ... , v+ 1). By Lemma 3. 2, we can find an increasing sequence of positive integers k p(p = 1,2, ... ) such that for each l~j~v+ 1, we have, in setting mp=nk,'
(5. 48) and
QuaSI-normal Famlhes of Meromorphic Functions
177
The sequence S :fmp (z) (p = 1, 2, ••• ) is also a Cj-sequence in D and has at most v nonCo-points in D. Among the v+1 points 1;,;(j=1,2,"',v+1), there is at least one 1;,; which is a Co-point of the sequence S. Consequently we can find a circle
Iz-1;,j I0 such that for each p~l we have (5.49)
in the circle
r I : Iz-1;,j Im2' ••• ,m. PHPl>P2' ••• ,p.)-filling domain of f(z). Consequently for any q sets Si E C (j= 1,2, ••• ,q) of points such that
there is an integer l~j~q such that the values of the function f(z) cover Sj with order of multiplicity less than mp strictly at least Pi+ 1 times. This theorem generalizes Theorems 3.14 and 3.15.
Proof. Let $T be the family of functions f(z) such that f(z) is meromorphic in D and that D is not a (ml>m2,···,m.PHPl>P2,···,p.)-filling domain of f (z). By Corollary 5. 5, $T is quasi-normal of order P at most in D. Next by Lemma 5. 12, there is a positve number
~
such that for each function f(z) E $T,
180
Normal
Fam~lies
of Meromorphic Functions
we have a(E ,P
Evidently this number
~
+ I,D;!) ~ J.L.
has the required property.
The second part of the conclusion of Theorem 5. 11 is an immediate consequence of the first part of that conclusion.
Theorem 5. 12. Given a domain D, a bounded closed set E of points belonging to D, a number 15>0 and M, TJ(O K,
then either the values of f(z) cover strictly at least p+ 1 times the domain Iwi < M or the values of
flY)
(z) cover strictly at least q + 1 times the domain TJ< Iw 1
M.
This result contradicts the assumption that the condit on in Theorem 5. 14 is satisfied.
5. 4. QUASI-NORMAL FAMILIES OF MEROM ORPHIC FUNCTIONS OF FINITE TOTAL ORDER In what follows, we first prove some preliminary theorems and finally introduce the notion of quasi-normal families of meromorphic functions of finite total order.
Normal Familtes of Meromorphic Functions
184
Lemma 5.13. Let S:fn(z)(n=1,2,···) be a Co-sequence of meromorphic functions in a domain D, such that the limit function Hz) of S, with respect to the spherical distance, is a meromorphic function in D. Let E be a bounded closed set of points belonging to D. Assume that Hz) is finite in E. Then we can find a positive integer N such that when 00
n~N,
fn(z) is finite in E and as n--i+
,fn (z) converges uniformly to f(z) in E. In a particular case, this lemma has been proved in the proof of Lemma 5. 8.
(see also the proof of Theorem 1. 1) In the general case, it is proved in the same way. However for the sake of completeness we give a proof. First of all, by Theorem 1. 2, as n--i+oo, fn (z) converges locally uniformly to Hz) in D, with respect to the spherical distance. Consequently to each point Zo E E corresponds a circle i (zo): I Z-Zo I O. Then for each j(1~j~m), there is a positive integer N j such that when n~Ni' we have If.(z) ,f(z)
I
O be a number such that 1Hz) IO and let S be a sequence of functions of gr. From S we can extract a subsequence SI which is a C 1-sequence in D and has at most P2 nonC o-
Qua&-normal Families of Meromorpkic Functions
197
points in D. Assume that SI has nonCo-points in D. Then by Lemma 5. 5, we can extract from SI a subsequence S2 which is either a Co-sequence in D or an irreducible CI-sequence in D. Consider the latter case and denote S2 by fn(z) (n = 1, 2, ••• ). Let E be the set of nonCo-points of S2 in D. E consists of a finite number of points zj(j=1,2"",m;1~m~p2)' In the domain DI=D-E, S2is a Co-sequence and hence has a limit function F (z) defined in D I , with respect to the spherical distance. For each point Zj let
r j:
I Z-Zj I
>h12
Let w be a value such that 0< Iw-aj IO), the inequality (6.31) becomes A
m(r,f.) ~ (1 -
r)"
and the series (6.32), neglecting a constant factor, becomes
(6.33)
NOTmal Famthes of MeTomoTplttc Functtons
236
L.: (1 -
lSi I)u+H".
i-I
In this way we deduce from Theorem 6. 6 a corollary which is more general than a generalization of Blaschke's theorem due to Montel [26J who ,instead of the condition (6. 33), assumes
logM(T,f.)
O, take a number
x'
=
o<e/2 and
+ U (~),).
x(1
(6.34)
then a positive integer no such that
for n?:no we have
T.
>
1
max (TO,TI)' -1--
T.
>
(6.35)
x 6'
Consider an integer n?:no and determine r by the relation
_1_ 1- T -
_1_11 1 - T.
+ U ( _11_ )6 1-
I ,
T.
which implies r n 0 is a constant and A=O+e)/O+2b»l. The series
is therefore convergent. This proves our assertion. Correspondingly in Theorem 6. 6. , we can remove the constant a> 1 in the series (6.32), provided that the function U (x) satisfies the condition (6.34) of normal growth.
Normal Families of Meromorphtc Functtons
238
6.2. GENERALIZATIONS OF BLOCH'S THEOREM AND SOME THEOREMS OF VALIRON The method of normal families is very useful for the study of the domains covered by the values of merom orphic functions. In the preceding chapters, we have already used that method in the proof of some particular covering theorems and in dealing with filling circles or filling domains. In this paragraph, we shall prove some general theorems in the case of holomorphic functions. Let us first recall Definition 2. 1 and the following theorem of Bloch (see Appendix A) :
Theorem 6.7. Let w=f(z) be a holomorphic function in the circle C: Izl 0 such that for each function f(z) E $T, there exists a circle Iw -Wo I
B. which is a simply covered image domain of f (z) for D.
Proof. Suppose that such a constant 13 does not exist. Then to each positive integer
n~l
corresponds a function fn(z)E$T, such that we can not find a
circle IW-Wo I
lin, which is a simply covered image domain of fn (z) for D. We are going to show that the sequence of functions f'n (z) (n = 1,2,,,,) converges locally uniformly to zero in D. In fact, consider a circle such that the circle Iz-zo I 0 be a number such that for each point aE').., the circle Iz-al~o belongs to D. Next take in the sense from Zo to z~, a finite number of points Zj (j = 0, 1, •.. , m) such that (6.40) By the result just obtained, the sequence fn,. (z) (k = 1,2, ... ) converges uniformly to 1 in the circle
r 0:
Iz -Zo I on the circles
IZ-Zj I =b
1
(6.47)
(j= 1, 2,,,, ,q), and therefore, by Rouche' s theorem, if
we denote by ""j~O the number of the zeros of the function g .. (z) -1 in the circle
r j ' we have
v.; =
v.;
(J = 1,2, .. ·,q).
Consequently if k~max(ko ,k~), then
2..:• v.; = 2..:• v.; ~ p , ;-1
;-1
incompatible with the condition 2' ). This contradiction proves that the sequence go, (z) (k = 1,2,,,,) is a Co-sequence in D and hence the family 0. By
(6.54), we can find a number R>r such that for each function qJ(z) E a, we have
in the domain Rl, fn,(z)-F(z) uniformly for r~lzl2, we can find a number P>Posuch that for each function g(w)E~, we have Ig(w) I>r for P< Iw 12, we can find P>Po such that for pO, 13>0 (a 1/,
Igj(z) 1< M (i = 1,2)
in the circle Iz-zol 0, = I,Z,---) (j = I,Z,---,p),m), = nt,
converge respectively to the limits /;'jEE 1 (j=I,Z,---,p) with
Is, - sll
~o (j,t = l,Z,---,p ;j=Fl).
(see Lemma 3. Z) One at least of the points /;'j(j=I,Z,---,p), say /;,j,ED-a. This leads to a contradicition in view of (7.8).
Corollary 7_ 4_ Let D, E,(i=I,Z), S,(i=I,Z) and p have the same meaning as in Corollary 7. 3. Then there exists a positive number b depending only on D,E;,SJi=l,Z) and p having the following property: If Hz) is a holomorphic function in D such that If(m)(zo) I::::;:; 1 (m
=
O,l,---,p - 1)
for a point Zo EEl and that
max If(z)1 ~b, %EB z
then the values of the function Hz) cover in generalized sense at least p times one of the sets S;O=l,Z).
Proof - Assume that the positive number b does not exist. Then for each positive integer n, there is a holomorphic function fn (z) in D such that
Normal Families of Meromorphic Functions
280
for a point z~n) EEl' that
and that the values of the function fn (z) do not cover in generalized sense at least p times either of the sets S.(j=1 ,2). Then as in the proof of Corollary 7.3, we find an increasing sequence of positive integers Ix (/. = 1 ,2, ... ) such that the sequence fl, (z) (A= 1,2, ••• ) converges locally uniformly to
00
in D-a, where a is a
set consisting of at most p -1 points of D, and that the sequence Z~I,) (/.= 1,2, ••• ) converges to a limit ~oE E 1 • By Definition 7. 1, we can find a circle
f:
Iz-~o
I ~r belonging to D such that the sets S.(j= 1,2) are uniformly bounded in f and that f -(/;'o)CD-a. Then on the circle Iz-~ol=r the sequence fl,(z) (/.=1,2, ••• ) converges uniformly to
fl,(z) - qi...z)
00,
=
and from the identity
(f1,(Z) -
P).(z)}
+ {P).(z)
- qi...z)},
where ,-I P).(z) =
~ h1,f~)(z~I'»(z
4-0
-
Z~I'»4,
•
we see that, when A is sufficiently large, the function fl, (z) -0 resulting from (7. 11). N ow consider a point Zo E D. Then one at least of the values a (zo) and b (zo)
is finite, say a (zo):;6oo. By Theorem 1. 1, we can find a circle ro: IZ-Zo I 0, Ic •• (z),a •• (z)
I >0,
Ib •• (z), c•• (z)
where 0 1/,
where band 11 are positive numbers. Since each of the three equations F.(z) =
00,
F.(z) = B.(z), F.(z) = C.(z)
has at most p distinct roots in r o, we can apply Lemma 7. 3 and conclude that the family {Ft(z) (k~ko)} is quasi-normal in ro of order p at most. This implies that the family
1
) F1 ( Z
= f.
(z) -
a. (z) (k ~ko)
)I
i
is also quasi-normal in ro of order p at most. Then by Lemma 7.1, the identity
f. .t (z) =
1
-I F (Z )
+ a. (z) .t
implies that the family {fn, (z) (k~ko)} is quasi-normal in ro of order p at most. Of course the same is true for the family {in, (z) (k = 1, 2, ••• ) }. Since Zo E D is arbitrary, we conclude, by Theorem 5.1, that the family {fn, (z) (k = 1,2, ••• )} is quasi-normal in D. Accordingly we can extract from fn, (z) (k = 1 ,2, ••• ) a subsequence f m• (z) (h =
1,2, ••• ) which is a C)-sequence in D. Then by Lemma 5. 5, we can extract
from f m • (z) (h = 1,2, ••• ) a subsequence fo, (z) (j = 1,2, ••• ) which is either a C osequence in D or an irreducible C)-sequence in D. Consider the latter case and let E be the set of the nonCo-points of the sequence fo,(z) (j=1,2,···) in D. Let f (z) be the limit function defined in D-E with respect to the spherical distance, of the sequence f.,(z) (j=1,2,···). Consider the seqeunces ao,(z), b.,(z) (j=1,2, ••• ) and their limit functions a (z), b (z) defined above. One at least of the two
298
Normal Families of Meromorp hic Functions
functions a(z).b(z) say a(z) is such that a(z)~f(z) in D-E. Assume that there are v points z,(j=1,2 ... • .v) belonging to E. Let
r ,: Iz-z, lO such that for each function f(z) E.7 we have If(z)1 ~),for z E E.
(7.28)
Proof. Assume that such a number A does not exist. Then to each positive integer n corresponds a function fn (z) E.7 such that
min If. 0 depending only on D. E).E 2 having the following property:1f f(z) is a holomorphic and univalent function in D satisfying the following conditions: 1 0 Hz) has a zero zoEE);
20 Ifl
(z~) I ~1 at a point z~E E 2;
then the values of f(z) cover in generalized sense the set Ho( Iw I 0 having the required property. Then to each positive integer n corresponds a function fn (z) E.7 such that the values of fn (z) do not cover in generalized sense the set Ho ( Iwi
0(k=1,2,···) and M.>0(k=1,2,···)
such that
lim r.
1-,)+=
and that the circles
=
0, ltm M. .-')+=
=+
00
r.: IZ-Zo I~r. (k = 1 ,2, ••• ) belong to D.
teger nl?:l such that
Next there is an integer n2>n] such that
T hen there is an in-
Normal Families of Meromorpliic Funct,ons
320
Continuing in this way, we get successively a sequence of integers nt (k = 1 , 2, ···;nt+l>nt) such that for k~l, we have
Consider a circle
f: IZ-Zo I~r belonging to D. Let
ko~l be an integer such that
rtO a number. Let f(z) be a meromorphic function in a domain D. A
r:
circle
Iz -Zo I O a number. Let
J}T
be a family of meromorphic functions in a do-
main D. If each function fez) of the family family
J}T
r.
J}T
has no (K ,e)-circle in D, then the
is Qm-normal in D.
Proof. Consider a point Zo of D and let r : Iz-zo I O a number. Let v?:O be an integer. If each function f(z) of (K , e)-circles in D, then
.'j}T
.'j}T
has at most v distinct
is Qm+l-normal of order vat most in D.
In the case v= 0, by Theorem 8. 17 the family
.'j}T
is Qm-normal in D, hence
the conclusion of Thoerm 8.18 holds. In the case v>O, Theorem 8. 18 is proved in the same way as for Theorem 5.3 in the case v>O. It is sufficient to replace in the proof of Theorem 5.3, the word "normal" by "Qm-normal", the term "Co-sequence" by "Cm-sequence", the term "CI-sequence" by "Cm+l-sequence" and the term "nonCo-point" by "nonCmpoint" , in order to get the proof of Theoerm 8. 18. Denoting by K' the condition of Theorem 8.27, namely "f(z) has at most v distinct (K ,e)-circles in D", then by Theorem 8.27, K' is a condition of Qm+lnormality of order vat most. The condition K' depends on m,K,e and v. We express this dependence in writing K' = £J(m,K ,t,v).
(8.6)
Now let Ko be a condition of Qo-normality. Let em>O (m = 0,1,2, ••• ) be a sequence of numbers and vm?:O (m=0,1,2,···) a sequence of integers. Then
is a condition of Q I-normality of order Vo at most;
is a condition of Q2-normality of order VI at most; in general, for m?:l,
is a condition of Qm-normality of order Vm-I at most. We may say that the sequence of conditions Km (m = 1 ,2, ••• ) is generated by the condition K 0 and the two sequences €m(m=0,1,2,···) and vm(m=0,1,2,···). N ext we give some generalizations of Theorems 3. 9,5. 4,3. 17 and Corollary 5. 7. For this we need the following definition:
Qm-normal
Fam~hes
of
Meromorph~c
333
Functions
Definition 8.13. Given a sequence of integers II=l,Im;?2 (m=2,3, .. ·) and a sequence of numbers Tlm
>0
tion Hz) in a domain D, a circle
(m = 1 , 2, ... ), consider a merom orphic func-
r : IZ-Zo Ip, a<argzO). Let D be a domain in a bounded domain ro n}
(9. 13)
cannot be contained in a finite number of circles of which the sum of the radii is less than
o.
From the sequence S: fn (z) (n = 1,2, ••. ) we can extract a subse-
quence S' : f n• (z) (k = 1,2, ••• ) which is a Cm-sequence in D. Let e be the set of nonCo-points of S' in D. In the domain Do=D-e, S' is a Co-sequence. As in the proof of Lemma 9. 5, we see that the limit function Hz) (with respect to the spherical distance) of S' can not be the constant
=,
and hence Hz) is a mero-
morphic function in Do· Now distinguish two cases: 1) The set ane is empty. Then aCD o· On a the function Hz) has at most a
finite num ber of poles
Zj
(j = 1 ,2, .•. ,h). Consider the circles
r j: IZ-Zj I n},
• r ,,) and it follows that U(U
Iln'
(9. 14)
can not be contained in a finite
j~l
number of circles of which the sum of the radii is less than 0/2. Since a. CD o , we can carryon the reasoning as in case 1), in replacing a by a. , 0 by 0/2 and (9. 13) by (9. 14), and get again a contradiction. This completes the proof of Lemma 9.9. Theorem 9. 11. Given a domain D, an integer m;;:?;O, a condition K of Qmnormality, a set ECD having the property Wm with respect to D, a positive function M (z) defined on E, a bounded closed set aCD and a positive number 0, we can find a positive number A(D,m,K ,E,M(z),a,o) such that if f(z) is a meromorphic function in D satisfying the conditions:
369
Apphcations of the Theory of Q .. -normal Families 1 ° Hz) satisfies the condition K in D, 2 0 1Hz) I~M(z) for zEE; then the set
{z Iz E O',lf(z) I
> A (D,m,K,E ,M(z) ,O',o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. To prove this theorem, it is sufficient to consider the family $T of all mero-
morphic functions in D satisfying the conditions 1° and 2°, and to apply Lemma 9.9.
Lemma 9. 10.
m~1 and v~O being two integers, let $T be a Qm-normal
of order v at most family of merom orphic functions in a domain D. Suppose that there is a set ECD satisfying the following conditions: 1° E has the property Wm_I.,+1 with respect to D. 2° At each point of E the family $T is uniformly bounded. Then for any bounded closed set aCD and any positive number 0, we can find a positive number o.(a,o) such that for each function Hz) E $T, the set
{z Iz E O',lf(z) I
> a(O',o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. Proof. Let aE D be a bounded closed set and 0 a positive number such that
a can not be contained in a finite number of circles of which the sum of the radii is less than
o.
Suppoes that we can not find a positive number o.(a,o) having the
required property. Then there is a sequence S:in (z) E $T (n = 1, 2, .. ·) such that for each positive integer n, the set Iln defined by (9. 13) can not be contained in a finite number of circles of which the sum of the radii is less then
o.
From S we
can extract a subsequence S' :in, (z) (k = 1,2, .. ·) which is a Cm-sequence in D and has at most v nonCm_l-points in D. Let e be the set of nonCm_l-points of S' in D. In the domain DI =D-e, S' is a Cm_I-sequence, and hence S' considered as a family, is Qm_I-normal in D I. On the other hand, there is a point zoEE Cm -l)nD I. Let
370
Normal
Fam~hes
of Meromorplnc Functwns
r : IZ-Zo IA(D,K,e,M,IT,o)}
can be contained in a finite number of circles of which the sum of the radii is less than O. To deduce this corollary from Theorem 9.13, it is sufficient to define a (0)collection {E} with respect to D, by considering each point of the set e to be a set E of the collection {E}.
Corollary 9.7. Given a domain D, a number h(OA(D,m,v,K,{E},M,u,6)} can be contained in a finite number of circles of which the sum of the radii is less than b.
Corollary 9.8. Given a domain D, an integer v;;?: 1 , a condition K of Q1normality of order vat most, a bounded closed set eCD and a positive number.." a positive number M ,a bounded closed set aCD and a positive number b, we can find a positive number A(D,v,K ,e,.."M ,a,b) such that if f(z) is a meromorphic function in D satisfying the conditions: 1 0 fCz) satisfies the condition K in D, 2 0 there exist v+l points zj Ee(j=1,2,···,v+I) such that
(9. 17) and that
then the set
376
Normal Families of
{zlz E 17,lf(z)1
Meromorph~c Funct~ons
>A(D,v,K,e,1/,M,I7,o)}
can be contained in a finite number of circles of which the sum of the radii is less than
o. In this corollary and the corollary below, it is tacitly assumed that there ex-
ist v+1 points zjEe(j=1,2,···,v+1) satisfying the condition (9.17). To deduce Corollary 9.8 from Theorem 9. 14, it is sufficient to define a (0, v+ 1 )-collection {E} with respect to D, by choosing the sets E to be the systems of points Zj E e (j = 1, 2, "', v+ 1) satisfying the condition (9. 17). (see the proof of Corollary 9.5)
Corollary 9.9. Given a domain D, an integer
v~l, a number h(O 1, f, (z) should have (w )m-circles (w
# 0,(0), when" is sufficiently large. We are going to show that this is really true. In fact, consider a circle T : 1z-iyo 1 /--] , ,,] =
1
w 1 1). Let us show that
(JogR +2rr) /0, the function f, (z) takes in T every value wEE.
Accordingly, by Definition 8. 13, for every wEE, the circle T is a (w) I-circle of f,(z). To prove this consider a point w=pe"EEO/R 0 such that for ">"zand wEE, each of the circles r;(j=I,2,···,Iz)is a (w)l-circle of the function f),(z) in r. Consequently r is a (w)z-circle of f),(z).
In general, basing upon Definition 8. 13, it is easy to prove by mathematical induction, the following assertion: Given an integer m~l, a circle r: Iz-iyoI O and 0O and satisfies the conditions ltm(/)a.)..u (X) x~o
= + 00,
ltm (/)a.)..V (X) x~+=
= O.
We denote by X=\If •. ,..u(Y) the inverse function of Y=.,,.,u(X). The function \If •. ".U (Y) is positive, decreasing and continuous for Y>O and satisfies the condi-
tions: LIm If'a.)..u(Y) y~+=
=
0,
l!mlf'a.)..u(Y) y~o
=+
00.
Theorem 1. Let m (t) be an increasing convex function for t
(2)
d. 0 ~ V no().
Proof. We have
~
~
.13}· ~ )..13}·-1 + 0 =
1
~
).)'~I
V (n)
.~J).d'-I + 0-).) vfn)/·
Hence there is an integer no~l such that
d. o ~).d. 0 - I
+ 0 - ) . ) V1-no()
which togather with the inequality dno-I~dno' gives (2).
Lemma 2. Let
dn(O~n~N .N~l) be a non-increasing finite sequence of
positive numbers. Let A and V (y) have the same meaning as in Lemma 1. Assume that
Under these conditions. there exists an integer l~no~N such that
d. o
>).d.
0
-I'
d. 0
1 > -V no().
(3)
Appendix A
396
Proof. We have
N
N
/~}.
> do + }'.I!}' + (1
N
'~1d.
-
N
> ).'~1d'-1 + (1 =
N 1 ).)'~1 V (n)'
1
N
).)'~1
-
'~1{M'-1 + (1
-).)
V (n)
vfn))'
Hence there is an integer l~no~N such that
1
d.>M.1 +(1-).)v-( o 0 no )
which togather with the inequality dno-1~dno' gives (3).
Lemma 3. Let h (s) be an increasing convex function for a ~s 1 do-)" + N
r
dy V(y)'
Hence by Lemma 2, there exists an integer l~no~N such that
d" >)"d, o- ] I d, 0
> V1-no()'
(1)
Then, as in case 1, by means of the convexity of the function h (s) and (1), we see that the three values Sl =00,-1 ,S2=00,' S3=00,+1 satisfy the conditions 1)-5) in Lemma 3, Now let us come back to the proof of Theorem 1. Put H=Ha,A.U and distinguish two cases according to
m(-H)~O
or -: m (to) - m (-
m+
to::--
to
+H
H)
>
m (to) H
(14)
•
The inequalities (12) and (14) yield
-t> 1 ~+ o m't- (to) 1 - A
r
(',J
~
(15)
Vex)·
Consider the increasing convex function met) for to~t eH-t:;',o.
In what follows, we first study the function f 2 (z), then the function fl (z) and finally the function f(z). Study of the function f 2 (z). By a theorem of Hadamard, the function m(t)
=
{ogM(e',f2)
(t
-
1)2,
(25)
where h a •• is defined by (16). Then (23) becomes
where 1j.'Q,'(Y) is defined by (16). Applying Corollary 1 to met), (I.,/., we conclude that there exist three numbers t j (j=1,2,3) satisfying the conditions 1)5) in Corollary 1. Put p =
e'·.
(26)
By the condition 1) in Corollary 1, we have (27)
Appendtx A
405
with
b
Taking a point
Zo
of the circle
1Z 1
=
= p,
(28)
h a ,;..
such that
(29) and introducing the number
(30)
we are going to show that N is real and
(31)
In fact consider a circle
r : 1 Z-Zo 1
).uI
N ~ ).a
=
a
aa
Consequently
Irp'
(-r) N
I
Z·
=-.l-.l
12 16'
(42)
Append~x
410
A
It follows that we have
(43)
with
(44)
in (C). By the condition 2) in Corollary 1, we have
n;;;:l being an integer, we have, by Cauchy's inequalities,
From (34) and (42), we have
p' -
p = pee" -
1)
> PO',
NO'>
a
2'
hence
(45)
Study of the function fJ (z). By (19) and (20), we have
(46)
411
Appendtx A and by (29) and the condition 4) in Corollary 1, we have
On the other hand from (38),(39),(29) and (36), we have in the circle (C),
Hence in the circle (C), we have
fl(zOe') =f2(zOe'){l +b(-r)},!b(-r)!
2 2
=
2e
131
1..) 6 < 2(2 e% + e < e%' a
X 12 X 16,
- 131< 1.. ~ Ib(r) 1 + 13 4e 16'
(53)
From (51), (53) and (39), we see that (52) holds. By (46) we have (54) Hence from (43) and (47), we have
+
zoe'f'l (zoe') _ eN, 1 WI ('I') Nfl(zo) 1+13'
Then from (44) and (47), we see easily that if we write
(55)
we have
(56)
in the circle (C). n~l being an integer, we deduce from (45) and (46),
(57)
with c =
a.
(58)
Appendix A
413
By (46) and (29), we have
(59)
(60) and
Then making use of the inequality
log (1 -
x)
x >- I - x
(0
2' Consequently if /:, is a point of (R), then on the boundary of (A), we have
IN .. + II
-
t; I
1 > 2'
hence by (69), we can apply Rouche' s theorem and conclude that the function A( .. ) -
t;
=
(N ..
+ II
-
t;)
+ A( .. )
has a zero ,=,(1:,) in the interior of (A), The function ,=,(1:,) is holomorphic and univalent in (R) and A { .. (t;) } = t;
in (R), It follows that in (R) the function u (t;) = .. (t;)
+
(71)
lOO
is hoi om orphic and univalent with Re{u(t;)} = Re{ .. (t;)}
Moreover by (70) we have
1,
1
a
-1--)e-to• r o' -
TO
•
(04)
432
Append,x A
We have
1
where
Zo -
So = 1 _
Z] Iz]1
To - T].. T] e' 0,
= 1_
I I So
< To·
Then by (04), we have M(TO,LogF l ) ;;?; log M(TO,F l ) ;;?; log IFl (so) I > logM(To,F)
> c,O + R +
ILogFl(O) I )e-I:.,o.
Hence by Corollary 8, we can find a number B>R and a holomorphic and univalent function q>l(W) in the circle Iw-woll(W)O-lzll) is holomorphic and univalent in the circle Iw-wolvlog(l-1'o) +logM(1'o'Ji)
+ c,(l + 2n- + R + vlog + c,(2n-v) (log
> c,(1
+ 2n- + R + vlog
1 ,o"gr• -1--)e- 1'0
1 " -1--)e-,ogr. - 1'0
1 " -1--)e-,ogr. -1'0
Then we complete the proof of Corollary 11 as in the proof of Corollary 9. As for Corollary 10, from Corollary 11 we can deduce the following corollary:
Corollary 12. Let F (z) be a function which is holomorphic and does not take the value zero in the circle
Iz I
(i~l, I ~
,
(16)
we have in (R),
(17)
By the properties 1° and 2° in Theorem 3, we see that for O~j~k, we have in (R) ,
(18)
where V j (/;) is a holomorphic function in (R) such that
Iv/s-) I < in (R),
r.
I,
+ klogN,
(19)
being a positive constant depending only on k. Consequently from
(18), (19) and (06), we have in (R),
(20) where V (I;,) is a holomorphic function in (R) such that
IV(S> 1< d (I, + klogN)
(21)
in (R). From (05),017),020) and the property lOin Theorem 3, in (R) we have
(22) where I (/;) is a holomorphic function in (R) such that
441
AppendIx A
11(S-)
I
(O)
-
II}
I + p'.
1}
(49)
We have 1 1 T(r,F) +log IF(O)I = T(r'[i)
(50)
and by (47), + + log T(p,F) ::;;'Iog (T(p,F)
+ log
al + 1 IF(O)I} ::;;'Iog T(p'[i)
+ log+ + log
al
+ log2. (51)
On the other hand, by Lemma 8 and (47), we have
+
1
+
1
O+T)/ogIF(O)I+Ploglog IF(O)I::;;'Pp
(2
1 + T)log 14'>(0) -
11
+ P log+
From (49)-(53), we deduce that for
1 T (r '-F)
(0) _ 11::;;' P2·
0