NORMAL FAMILIES AND SHARED VALUES OF HOLOMORPHIC FUNCTIONS

Let F be a family of holomorphic functions in a domain D, k be a positive integer, a, b（≠0）, c（≠0） and d be finite complex numbers. If, for each f∈F, all zeros of f-d have multiplicity at least k, f^（k） = a whenever f=0, and f=c whenever f^（k） = b, then F is normal in D. This result extends the well-known normality criterion of Miranda and improves some results due to Chen-Fang, Pang and Xu. Some examples are provided to show that our result is sharp.

Normal Families of Holomorphic Functions Concerning Shared a Polynomial

In this paper,we study normal families of holomorphic function concerning shared a polynomial.Let F be a family of holomorphic functions in a domain D,k(2)be a positive integer,K be a positive number andα(z)be a polynomial of degree p(p 1).For each f∈F and z∈D,if f and f sharedα(z)CM and|f(k)(z)|K whenever f(z)-α(z)=0 in D, then F is normal in D.

Normality Criteria for Families of Holomorphic Functions

Let f be a holomorphic function on a domain D (?) C, and let a be a finite complex number. We denote by Ef(α) = {z∈ D : f(z) = a, ignoring multiplicity} the set of all distinct α-points of f. Let F be a family of holomorphic functions on D. If there exist three finite values a, b(≠ 0, a) and c(≠0) such that for every f ∈ F, Ef(0) (?) Ef'(a) and Ef'(b)(?) Ef(c), then F is a normal family on D.

NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS SHARING A HOLOMORPHIC FUNCTION AND THE CONVERSE OF THE BLOCH PRINCIPLE

In this paper, we investigate normal families of meromorphic functions, prove some theorems of normal families sharing a holomorphic function, and give a counterex- ample to the converse of the Bloch principle based on the theorems.

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result.Let F be a family of holomorphic functions on a domain D C,all of whose zeros have multiplicity at least k,where k≥2 is an integer.And let h(z)■0 be a holomorphic function on D.Assume also that the following two conditions hold for every f ∈ F:(a) f(z)=0■|f(k) (z)|<|h(z)|;(b)f(k)(z)≠h(z).Then F is normal on D.

Hyperbolic Metric and Normal Family

In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.

Some Results on Unicity of Entire Functions Related to Normal Families

This paper studied the connection between normal family and unicity, and proved some results on unicity of entire functions. Mostly, it was proved: Let f be a nonconstant entire function, and let a, c be two nonzero complex numbers. If E(a,f)=E(a,f′), and f″(z)=c whenever f′(z)=a, then f(z)=Ae~ cza +ac-a^2c. The proof uses the theory of normal families in an essential way.

涉及不取零点的亚纯函数族的正规性

主要讨论了一类涉及例外函数是全纯函数的亚纯函数正规定则。通过利用Pang-Zalcman引理,在前人研究的基础上得到一类新的正规定则。

有关全纯函数正规族的一个结果

设F是单位圆盘Δ上的全纯函数族,a和b是两个判别的有穷复数,c为任意非零复数,若对任一f∈F,有-f(0)-f′(a),-f(c)-f′(b),则F在单位圆盘Δ上正规.

全纯函数的一个正规族(英文)

设F为单位圆盘Δ上的一个全纯函数族,M,N为两个正实数.如果对于任意的f∈F,f的零点重级≥m,且f(z)=0|f^((m))(z)|≤M,f^((m))(z)=1|f(z)|≥N,则F在Δ上正规.

关于Ошкин的一个定理

本文研究了全纯函数族的正规性,得到如下结果: 定理1 设为区域D上的一族全纯函数,n,k(k≥2)为两正整数,占为非零有穷复数,a_1(z),a_2(z),…,a_k(z)均在D内全纯。若对中每一个函数f(z)均有:(1)f(z)的零点重数≥k;(2)f^n(z){f^((k))(z)+a_1(z)f^((k-1))(z)+…a_k(z)f(z)}≠b,则在D正规。

A Criterion of Normality Based on a Single Holomorphic Function

Let F be a family of functions holomorphic on a domain D C C. Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k- 1, such that h（z） has no common zeros with any f∈F. Assume also that the following two conditions hold for every f ∈F ：（a） f（z） = 0 == f （z） = h（z）; and （b） y（z） = h（z） == ｜f（k）（z）｜ ≤ c, where c is a constant.Then F is normal on D.

亚纯函数的正规族(英文)

关于正规族的 Hayman猜测目前已完全证实 ,本文考虑把 Hayman猜测中的 f′换为一般的f( k) ,得到一个更为一般的结果 ,由此改进和推广了陈怀惠 ,顾永兴 ,华歆厚 ,庞学诚与 W.

与分担值相关的正规族

设F为区域D上的亚纯函数族,a和b都是不为零的两个有穷复数(a/b不是正整数),若每个f∈F,f(z)=a(?)f^((k))(z)=a,且f-a的零点重级至少为k,当f^((k))(z)=b时有|f(z)-a|≥ε,其中ε为正数.则F在区域D内正规.

分担值与正规族

本文得到如下结果 :设 F为区域 G上的全纯函数族 .若族 F中每个函数 f和其导数f′以 1为 CM公共值 ,且 N( r,1 /f′) <λT( r,f) ,0 <λ<1 /4,则 F在

涉及分担值的正规族

主要讨论分担值与正规族的问题 ,改变 Schwick

与其导数具有分担值的全纯函数的正规族

证明了一个区域上的全纯函数族是正规的，如果任意的两个相互判别的有限非零值为族中任意函数及其一阶导数在该区域中的ＣＭ分担值。

涉及微分多项式的亚纯函数正规族(英文)

给出了一个一般性的正规定则.设F为区域D上的一个亚纯函数族,h0,a0,a1,…,am-1为区域D上的全纯函数.如果对于任意的f∈F,f的极点重数≥2,f的零点重数≥m+2,且L(f)(z)=f(m)(z)+am-1(z)f(m-1)(z)+…+a1(z)f′(z)+a0(z)f(z)≠h(z)z∈D则F在区域D上正规.

球面导数，高阶导数与正现族

本文应用陈怀惠和顾永兴关于Ｚａｌｃｍａｎ不正规性条件的改进结果，推广和加强了Ｌａｐｐａｎ的一个正规定则．Ｌａｐｐａｎ证明：若亚纯函数族中的所有函数的球面导数的幂方（大于２）在紧集上的积分一致有界，则该族是正则的．本文证明，把积分限制在函数值的模小于给定常数的子集上，结论仍然成立．同时，用高阶导数的积分替代球面导数的积分，得到十分一般的结果．另外对幂方为２的情形也进行了讨论．

分担连续函数的全纯函数正规族

研究了分担连续函数的全纯函数族的正规性问题,推广了一些已有的结论.设F为定义在区域D上的全纯函数族,h1,h2为两个连续函数满足对_z∈D有h1(z)≠h2(z),并设k≥2为正整数.若f∈F,有f(z)=hi(z)=〉|f(k)(z)|≤|hi(z)|,i=1,2,则F为D上的正规族;并举例说明了k=1时,结论不成立.此外,还将分担值条件用拓扑度条件代替得到了一个涉及拓扑度条件的全纯函数族正规定则.