A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omit a Function II

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of functions holomorphic on a domain D ? ?, all of whose zeros have multiplicity at least k, where k ?? 2 is an integer. Let h(z) ? 0 and ?? be a meromorphic function on D. Assume that the following two conditions hold for every f ?? F: $$ \begin{gathered} (a)f(z) = 0 \Rightarrow |f^{(k)} (z)| < |h(z)|. \hfill \\ (b)f^{(k)} (z) \ne h(z). \hfill \\ \end{gathered} $$ Then F is normal on D.     

A Normal Criterion Concerning Omitted Holomorphic Function

In this paper, we continue to discuss the normality concerning omitted holomorphic function and get the following result. Let F be a family of meromorphic functions on a domain D, k ≥ 4 be a positive integer, and let a(z) and b(z) be two holomorphic functions on D, where a(z) 0 and f(z) ≠ ∞ whenever a(z)=0. If for any f ∈ F, f'(z) -a(z)f^k(z) ≠ b(z), then F is normal on D.     

函数与其导数具有公共值的全纯函数族的正规性

设F为区域G上的全纯函数族, a,b(≠0)为两有穷复数,n为正整数,本文推广了Miranda定则,证明了:若对任意的f∈F,(a,b)为f与f(n)在G上的IM分担数组,且当f=a时, f'=f(n+1)=b,则F在G中正规.     

涉及分担超平面的正规定则

在本文中,作者继续讨论涉及分担超平面的全纯曲线的正规性,得到了如下结果:设F是一族从区域D?C到PN?上的全纯曲线,Hj={x∈PN?:(x,αj)=0}是PN?中处于一般位置的超平面,这里αj=(aj0,…,ajN)T且aj0≠0,j=1,2,…,2N+1.若对于任意的f∈F,满足下列两个条件:(i)如果f(z)∈H...     

涉及分担函数的正规定则

设k为正整数,M为正数;F为区域D内的亚纯函数族,且其零点重级至少为k;h为D内的亚纯函数(h(z)≠0,∞),且h(z)的极点重级至多为k.若对任意给定的函数f∈F,f与f~((k))分担0,且f~((k))(z)-h(z)=0?|f(z)|≥M,则F在D内正规.     

Functions Holomorphic along Holomorphic Vector Fields

The main result of the paper is the following generalization of Forelli’s theorem (Math. Scand. 41:358–364, 1977): Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with eigenvalues whose ratios are positive reals. Then any function φ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary. K.T. Kim and G. Schmalz were supported by the Scientific visits to Korea program of the AAS and KOSEF. E. Poletsky was supported by NSF Grant DMS-0500880. G. Schmalz gratefully acknowledges support and hospitality of the Max-Planck-Institut für Mathematik Bonn.     

公共值与正规定则

设F为区域G上的全纯函数族,a,b为两个判别的有穷复数．本文证明了如果对任意的f∈F,a;b为f与f'在G上的IM公共值,则F在G上正规．     

全纯函数族的正规定则

Let f be a holomorphic function on a domain D Lontaiu in C, and let a be a finite complex number. We denote by -↑Ef/(a) = {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(≠ O,α) and c(≠ O) such that for every f ∈F, -↑Ef′/(0) Lontain in -↑Ef(α） and -↑Ef′ (b) Lontain in -↑Ef(c), then F is a normal family on D.     

关于分担多项式的全纯函数的正规族(英文)

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.     

涉及分担函数与分担值的正规定则

研究了涉及分担函数的正规定则,证明了:设F为定义在区域D内的一族亚纯函数,n,k是两个正整数,满足n≥k+3.如果对于F中任意一个函数f,(fn)(k)-z至多有一个不同的零点,则F在D内正规.此结论说明在(fn)(k)具有不动点的情形下,1990年杨乐在Notre Dame大学举行的学术会议上提出的断言仍然成立.     

代数体函数的正规族定理

设F是定义在球面V的子集D上的一族代数体函数,若F中所有f的分支点之和为有限个，且存在三个固定的复值a₁,a₂,a₃使得对每个f都有∑³_k=1n(D,a_k, f)≤1,则F在D内正规.     

涉及导数的一类全纯函数的正规族

主要使用Zalcman引理来研究全纯函数的正规族,得到了如下的结论:令F为|z|<1内的一族全纯函数,n是一个正整数,a,b是两个复数且满足a≠0,∞,b≠∞.若F满足:Ⅰ)■f∈F,如f有零点,则f的零点重级大于等于3;和Ⅱ)当n≥4时,对F的每一对函数G和H,G″-aG~(n,)与H″-aH~n分担b.则F在|z|<1内正规.     

一类亚纯函数的正规定则

该文研究了一类亚纯函数的一般性的正规定则.其结果推广了以前与之相关的一系列结果.     

多复变全纯映射正规族关于连续移动超平面的一个充要条件

证明了从C~n的一个区域到P~N(C)中的多复变全纯映射族为正规族的一个充要条件.这个充要条件与P~N(C)中的逐点处于一般位置的连续移动超平面族有关.证明的主要方法来自于微分方程.     

全纯函数的正规族

本文证明了两个正规族定理,分别改进了张国明、孙伟和庞学诚的一个结果和徐焱的一个结果.     

亚纯函数族涉及微分单项式分担移动靶的正规定则

设a(z)是一个没有零点的整函数,k≥3是个整数,F是区域D上的亚纯函数族,对每一个f∈F至少有k重零点和2重极点.若对每一对f,g∈F有ff(k)与gg(k)IM分担a(z),则F在区域D内正规.     

Normality and shared-value of derivatives

We show that a family F of functions holomorphic in a domain is normal, if for all ? ∈ F the derivative ?′ and ? share the value a(≠0) and if the set of values that ? _(k+1) takes the zeros of ??a, is bounded. As an application we obtain some new uniqueness theorems of entire functions which improve the former results [Amer. H.H. Al-khaladi (2000). On a question of Yi-Yang. J. Shandong Univ., 35(3), 162–167; G. Jank, E. Mues and L. Volkmann (1986 Jank, G., Mues, E. and Volkmann, L. 1986. Meromorphic functionen die mit ihrer ersten und zweiten Ableitung einen endlichen wert teilen. Complex Variables, 6: 51–71. [Taylor & Francis Online] , [Google Scholar]). Meromorphic functionen die mit ihrer ersten und zweiten Ableitung einen endlichen wert teilen. Complex Variables, 6, 51–71; P. Li and C.C. Yang (2001). Uniqueness theorem on entire functions and their derivatives. J. Math. Anal. Appl., 253, 50–57; W. Lin (2001). Uniqueness theorem and normal criteria with sharing array. Math. Theory Appl., 21, 9–16 (Chinese); H.X. Yi and C.C. Yang (1995). Uniqueness Theory of Meromorphic Functions, Pure and Applied Math. Monographs No. 32. Science Press, Beiling; H. Zhong (1995). Entire functions that share one values with their derivatives. Kodai Math. J., 18, 250–259, etc].     

Montel正规定则的推广

用简单的方法证明了全纯函数族的一个正规定则,推广了Montel正规定则.设F为区域D上的一个全纯函数族,其零点是重级的,a为有限复数.如果f,g∈F,有(f-1)f′与(g-1)g′分担a,则F正规.     

正则解析半群与正则Cosine函数

在本文中，我们的主要结果是：若Ａ是一个正则解析半群的生成元，则Ａ也一定是某正则Ｃｏｓｉｎｅ函数的生成元．