Sharing set and normal function of holomorphic functions Текст научной статьи по специальности «Математика»

2000 MSC 32A15, 32A20, 32A22, 32H30

SHARING SET AND NORMAL FUNCTION OF HOLOMORPHIC FUNCTIONS Jun-Feng Xu

Department of Mathematics, Wuyi University,

Jiangmen, Guangdong 529020, China, e-mail: lvfeng@mail.sdu.edu.cn

Abstract. In this paper, we use the idea of sharing set to prove: Let F be a family of holomorphic functions in the unit disc, a1 and a2 be two distinct finite numbers and a1 + a2 = 0. If for any f G F, Ef (S) = Ef (S), S = {ai, a2}, in the unit disc, then f is an a-normal function.

Keywords: entire functions, uniqueness, Nevanlinna theory, normal family.

1 Introduction and main results

Let D be a domain in C and let F be a family of meromorphic functions defined in D. The family F is said to be normal in D, in the sense of Montel, if each sequence {fn} C F contains a subsequence {fn.} that converges, spherically locally uniformly in D, to a meromorphic function or to TO.(see. [10])

In this paper, we assume that /, g are two meromorphic functions on D and Si, S2 are two sets. We denote Ef(S\) C Eg(S2) by /(z) G Si => g(z) G S2. If Ef(S\) = Eg(S2), we denote this condition by f (z) G S1 ^ g(z) G S2. Similarly, if Ef (S1) = Eg(S2), we denote this condition by f (z) G S1 ^ g(z) G S2. If the set S has only one element, say a, we denote f (z) G S by f (z) = a (see [15]).

Schwick[14] was the first to draw a connection between values shared by functions in F (and their derivatives) and the normality of the family F. Specially, he showed that if there exist three distinct complex numbers a1, a2, a3 such that f and f' share aj(j = 1, 2, 3) in D for each f G F, then F is normal in D. Pang and Zalcman [9] extended this result as follows.

Theorem A. Let F be a family of meromorphic functions in a domain D, and let a, b, c, d be complex numbers such that c = a and d = b. If for each f G F we have f (z) = a ^ f '(z) = b and f (z) = c ^ f '(z) = d, then F is normal in D.

Definition 1.1 (see. [6, 7]) A meromorphic function f is a normal function in the unit disc D if and only if there exists a constant C(f) (which depends on f) such that

(1 - |z|'2)/«(;) < C(f),

where f N(z) = |g'(z)|/(1 + |g(z)|2) is the spherical derivative of f.

In 2000, X.C. Pang [8] considered the normal function by using the condition of share values.

Theorem B. Let F be a family of meromrophic functions in the unit disc, a1, a2 and a3 be three distinct finite numbers. If for any f G F,

Ef(ai) = Ef(ai), i= 1,2,3,

The author was supported by the NSF of China (10771121), the NSF of Guangdong Province (9452902001003278) and Excellent Young Fund of Department of Education of Guangdong (LYM08097).

in the unit disc, then there exists a positive M, such that for every f G F, we have

(1 - |z|2)f«(z) < M,

where M depends on a1, a2 and a3.

In fact, from the proof of Theorem B, one can get the following corollary.

Corollary 1.2 Let F be a family of holomorphic functions in the unit disc, a1 and a2 be two distinct finite numbers. If for any f G F,

Ef(ai) = Eft(cii), '¿=1,2,

in the unit disc, then the conclusion of Theorem B holds.

Recently, there exist a lot of studies in using the shared set to obtain the normal family(see. [2, 4, 5]). X.J. Liu obtained a normal function by using the share set S = {a1, a2, a3} corresponding Theorem B. Naturally, we ask whether there exists a normal function by using the shared set S = {a1,a2} corresponding to Corolllary 1.2? In this paper, we study the question and get the following result.

Theorem 1.3 Let F be a family of holomorphic functions in the unit disc, a1 and a2 be two distinct finite numbers and a1 + a2 = 0. If for any f G F,

Ef (S ) = Ef, (S), S = {a1,a2},

in the unit disc, then there exists a positive M, such that for every f G F, we have

(1 - |z|2)f«(z) < M,

where M depends on S.

In the following, we give a example to show the condition a1 + a2 = 0 is necessary.

Example 1.4 ([5]) Let S = {-1,1}. Set F = {fn(z) : n = 2, 3, 4,...}, where

n + 1 n — 1

/.(.-) = ^ + — e- D = {:: |;|<1}.

Then, for any fn G F, we have

n2[fn2(z) -1] = fn2(z) -1.

Thus fn and fn share S CM, but fn is not a normal function in D.

From Case 1 in the proof of Theorem 1.3, we can easily get the following corollary.

Corollary 1.5 Let F be a family of functions holomorphic in a domain D, let a be a nonzero

finite complex numbers. If for all f G F, f and f' share S = {0, a} IM, then the conclusion of

the theorem 1.3 holds.

The following example shows that it is necessary that the complex numbers a is finite.

Example 1.6 Let S = {0, to}. Set F = {enz : n =1, 2,...} in the unite disc A, thus fn = enz and f'n = nenz share S, but f is not a normal function in A.

Definition 1.7 ([11]) Given 0 < a < to, if there exists a constant Ca(f) such that

(1 - M2)“7«(z) < Ca(f), for each z G D, we say that f is an a-normal function in D.

a-normal functions may be viewed as the generalizations of normal functions. If we denote by N the class of the normal functions in D and denote by Na the class of the a-normal functions in D, it is obvious that

Nai C N C Na2

for 0 < a1 < 1 < a2 < to. The above inclusion relations are strict(see.[12]). Similarly, we can get the following generalized result.

Theorem 1.8 Let a > 1, and let F be a family of holomorphic functions in the unit disc, a1

and a2 be two distinct finite numbers and a1 + a2 = 0. If for any f G F,

Ef (S ) = Ef, (S), S = {a1,a2},

in the unit disc, then there exists a positive M, such that for every f G F, we have

(1 - |;|2)7«(z) < m,

where M depends on S.

2 Lemmas

Lemma 2.1 ([9]) Let F be a family of functions meromorphic on the unit disc, all of whose zeros have multiplicity at least k, and suppose that there exists A > 1 such that |f (k)(z)| < A whenever f G F and f (z) = 0, f G F. Then if F is not normal, then there exist, for each 0 < A < k,

(a) a number 0 < r < 1;

(b) points zn, zn < 1;

(c) functions fn G F, and

(d) positive number pn 0 such that p-Afn(zn + an£) = gn(£) ^ g(£) locally uniformly, where g is a nonconstant meromorphic function on C such that g«(£) < g«(0) = A + 1.

The normal lemma is for a-normal functions corresponding to Lemma 2.1.

Lemma 2.2 Let F be a family of functions meromorphic on the unit disc, all of whose zeros have multiplicity at least k, and suppose that there exists A > 1 such that |f (k)(z)| < A whenever f G F and f (z) = 0, f G F. Then if F is not an a-normal function, then there exist, for each 0 < A < k and 1 < a < to, there exist a sequence of points {zn} in D and a sequence of positive numbers {pn} such that |zn| ^ 1, pn ^ 0, and the sequence of functions

{gn(()} = PnXf (zn + (1 - |zn|2)apnC)

converges spherically and locally uniformly to a non-constant Yosida function in the (-plane.

Remark. The case 0 < A < k is first proved by Chen and Wulan, see [12, 13] for a detail. We can prove the above lemma by the similar method with [13].

3 Proof of Theorem 1.8

Suppose, to the contrary, that we can find |zn| < 1 and fn G F such that

gn(z) = fn(zn + (1 - |zn|2)az) (3.1)

satisfy

lim g«(0) = lim (1 - |zn|2)af«(zn) = to.

n—n—

Hence {gn(z)} is not normal in the unit. By Lemma 2.1, we can find the positive number r, 0 < r < 1; the complex numbers (n, |Zn| < 1; pn ^ 0+ and gn G F such that

Gn(C) = gn(Cn + Pn() = fn(zn + (1 - |zn|2)aCn + (1 - |zn|2)aPnC)

locally uniformly to a nonconstant entire function G(Z) on C.

We know G is a nonconstant entire function. Without loss of generality, we can assume that G - a1 has zeros in C. Let Z0 is a zero of G - a1. Consider the family

M = №(c); №

We claim H is not normal at (0. In fact, G(Z0) = a1 and G(Z) ^ a1. From (3.1) and Hurwitz’s Theorem, there exist (n, (n ^ Z0 and Gn((n) = a1. Then Hn(Zn) = 0. However, there exists a positive number 5 such that As = {z G D : 0 < |Z - Z0| < 5} C D and G(Z) = a1 in As. Thus for each Z G As, Gn(() = a1 (for n sufficiently large). Therefore for each Z G As, we have H(() = to. Thus we have proved that H is not normal at (0.

Noting that

Hn(Z) = 0 ^ H’n(Z) = a1 or a2, and using the Lemma 2.1 again we can find Tn ^ t0, nn ^ 0 and Hn G H such that

n Hn(Tn + Vn£,) Gn(Tn + pnC) - a1

fn{^) — —

Vn Vn

_ fn (Zn + (1 — | Zn | 2)a C»i + (1 — \zn\2)a Pni^n + VnQ) — al

(1 \Zn\'^')a PnVn

locally uniformly convergence to F(C) on C, where F is a nonconstant entire function such that F«(£) < F«(0) = M. In particular p(F) < 1.

We claim that

(1) F only has finitely many zeros.

(2) F(£) = 0 ^ F'(£) = a1 or a2.

We first prove Claim (1). Suppose Z0 is a zero of G(Z) - a1 with multiplicity k. If F(£) has infinitely many zeros, then there exist k +1 distinct points Cj (j = 1, ■ ■ ■ ,k +1) satisfying F(Cj) = 0 (j = 1, ■ ■ ■ ,k +1). Noting that F(£) ^ 0, by Hurwitz’s Theorem, there exists N, if n> N, we have Fn(Cjn) = 0 (j = 1, ■ ■ ■ ,k + 1) and Gn(Tn + nnCjn) - a1 = 0. We have

lim Zn + nnCjn = C0, (j =1, ■ ■ ■ ,k +1)

n—tt

then Z0 is a zero of G(Z) - a1 with multiplicity at least k + 1, which is a contradiction. Thus we have proved Claim (1).

Next we prove Claim (2). Suppose that F(C0) = 0, then by Hurwitz’s Theorem, there exist Cn, Cn ^ C0, such that (for n sufficiently large)

p \ _ fn(zn + (1 — |z«|2)“Cra + (1 — \~n\2)aPn{rn + 1]nCn)) ~ al _ p

“ (1 - |^|2)«Pn^ “

Thus fn(zn + (1 - |zn|2)“(n + (1 - |zn|2)aPn(Tn + VnCn)) = «1. By the assumption, we have fn(zn + (1 — |zn|2)°Cn + (1 — |zn|2)aPn(Tn + VnO) = a1 or «2,

hence

F'(C0) = lim fn (zn + (1 — |zn|2)°Cn + (1 — |zn|2)apn (Tn + VnCn)) = a1 or «2-

n—

Thus we prove F(C) = 0 ^ F'(£) = a1 or a2.

In the following, we will prove F'(C) = a1 or a2 ^ F(C) = 0.

Suppose that F'(Co) = a1. Obviously F' ^ a1, for otherwise FB(0) < |F'(0)| = |a1| < M, which is a contradiction. Then by Hurwitz’s Theorem, there exist Cn, Cn ^ Co, such that (for n sufficiently large)

Fn(C'n) = fn(zn + (1 — |zn|2)“Cn + (1 — |zn|2)aPn(Tn + VnCn)) = a1-

It follows that Fn(Cn) = fn(zn + (1 - |zn|2)“Cn + (1 - |zn|2)aPn(Tn + VnCn)) = a1 or a2.

If there exists a positive integer N, for each n > N, we have

fn(zn + (1 - |zn|2)°Cn + (1 - |zn|2)aPn(Tn + VnCa)) = a2 -

Then

fn(zn + (1 - |zn|2)“Cn + (1 - |zn|2)“Pn(Tn + VnCn)) - a1

f {£o) = hm --------------------------- ---:——-------------------------------= oo,

ra^°° (1 - \zn\ )aPnVn

it contradicts with F'(C0) = a1. Hence there exists a subsequence of {fn}(which, renumbering, we continue to denote by {fn}) satisfying that

fn(zn + (1 - |zn|2)°Cn + (1 - |zn|2)aPn(Tn + VnCa)) = a1-

Thus we derive

\ V fn(zn + (1 - |zn|2)“Cn + (1 - |zn|2)aPn(Tn + VnCn)) - a1 n

■fr (to) — hm -------------------------—---j——-------------------------------— 0,

ra^°° (1 — |zra| )“PnVn

which implies F' = a ^ F = 0. Similarly, we can get F' = a2 ^ F = 0. Hence we have proved claim (2).

Since p(F') = p(F) < 1, then by the Nevanlinna’s second fundamental theorem,

T(r, F') < N(r, — ---------) + N(r, —^----------) + S(r, F!)

F' - a1 F' - a2

< N(r, ---------) + N(r, ------------) + 0(log r) (3.2)

F — a1 F — a2

< N(r, -^) + O(logr)

F

From Claim (1), we get N(r, -^) = O(logr). Thus T(r,F') = O(logr), it is clear that F is a polynomial.

In the following, we consider two cases:

Case 1: a1a2 = 0. Without loss of generality we assume a1 = 0. We know that F' has zeros, then F has multiple zeros. We assume deg(F) = n, then T(r, F') = (n - 1)logr and S(r, F') = O(1). By (3.2) we get

T(r, F') = (n — l)logr < N(r, -^) + 0(1) < (n — l)logr

F

Thus we derive that F only has one multiple zeros with multiplicity 2 and F' only has one zero with multiplicity 1, which yields that n =2. Set F' = B(C - Co), then F = (B/2)(C - Co)2, which contradicts with F' = a2 ^ F = 0. This completes the proof of Case 1.

Case 2: a1a2 = 0. We first prove F = 0 ^ F' = a1 or a2. From a1a2 = 0, we get F = 0 ^ F' = a1 or a2. Thus we only need to prove F' = a1 or a2 ^ F = 0.

Suppose Co is a zero of F' - a1 with multiplicity m. By Rouche theorem, there exist m sequences {Cin}(i =1, 2 ■ ■ ■ ,m) on D^/2 = {C : |C - C0| < ^/2} such that Fn(Cin) = a1. Then

fn(zn + (1 - |zn|2)°Cn + (1 - |zn|2)aPn(Tn + VnCm)) = F'n(C,in) = a1 (i = 1) 2, ' ' ' j m)-

By f and f' share {a1; a2} CM, we get f' - a1 only has simple zeros. That is Cin = Cjn(1 < i = j < m). We obtain

fn(zn + (1 - |zn|2)“Cn + (1 - |zn|2)aPn(Tn + VnCin)) = a1 or a2 (i = 1, 2, ■ ■ ■ ,m).

We claim that there exist infinitely many n satisfying

fn(zn + (1 - |zn|2)°Cn + (1 - |zn|2)aPn(Tn + VnCin)) = a1 (i = 1) 2, ■ ■ ■ ,m)- (3.3)

Otherwise we may assume that for all n, there exist j G (1,... ,m) satisfying

fn(zn + (1 - |zn|2)°Cn + (1 - |zn|2)apn(Tn + nnCin)) = a2•

We take a fixed number l G (1, •••,m) satisfying (for infinitely many n)

fn(zn + (1 - |zn|2)°Cn + (1 - |zn|2)apn(Tn + nnCin)) = a2.

Hence

Pic \ y fn(~n + (1 — |zn|2)aCii + (1 — \zn\2)aPnijn + VnCin)) — Cl 1

F(£ 0) = hm

n—^ (1 - |zn|2)apn^n

a2 - a1

= hm —----------:—— -------= oo,

n—TO (1 - |zn| )apn^n

which contradicts with F'(C0) = a1. This proves (3.3). Therefore

Fn(Cin) = 0, (i = 1, 2, ■ ■ ■ , m)

and Cin = Cjn (1 < i = j < m). As n ^ to, we get Co is a zero of F with multiplicity at least m. This proves F' = a1 ^ F = 0. Similarly we can get F' = a2 ^ F = 0. Thus we have proved

F = 0 ^ F' = a1 or a2.

From this we know F — a and F — a2 only have simple zeros. Suppose that deg(F) = n, then n = 2(n — 1) and n = 2. Set F = A(£ — ^1)(C — £2), then F; = A(2£ — ^1 — £2).

Without loss of generality, we assume that F/(^1) = a1 and F'(£2) = a2, we get a1 + a2 = 0. It is a contradiction.

Thus we complete the proof of Theorem 1.3.

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РАЗДЕЛЕННОЕ МНОЖЕСТВО И НОРМАЛЬНАЯ ФУНКЦИЯ ГОЛОМОРФНЫХ ОТОБРАЖЕНИЙ

Жан Фенг Ксю Вууй Университет,

Цзянмэнь, Гуангдонг 529020, Китай, e-mail: lvfeng@mail.sdu.edu.cn

Аннотация. В работе идея разделенного множества применяется к описанию нормальных функций для семейства мероморфных функций в единичном круге.

Ключевые слова: целая функция, единственность, теория Неванлинна.