正规函数与正规族

研究了正规函数与正规族之间的联系,利用分担集合的方法得到一族满足特殊性质p的正规函数,同时利用分担值的方法得到了一个较已有结论更为一般的结果。     

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.     

Normality concerning shared values

Let be a family of meromorphic functions in a plane domain D, and a and b be finite non-zero complex values such that . If for and , then is normal. We also construct a non-normal family of meromorphic functions in the unit disk Δ={|z|<1} such that for every and in Δ, where m is a given positive integer. This answers Problem 5.1 in the works of Gu, Pang and Fang. This work was supported by National Natural Science Foundation of China (Grant Nos. 10671093, 10871094) and the Natural Science Foundation of Universities of Jiangsu Province of China (Grant No. 08KJB110001), the Qing Lan Project of Jiangsu, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry     

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 counterexample to the converse of the Bloch principle based on the theorems.     

On normal families of meromorphic functions

T. Qian In this paper, we prove two theorems on normal families of meromorphic functions, which improve a few of results from several authors. Copyright © 2015 John Wiley & Sons, Ltd.     

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 counterexample to the converse of the Bloch principle based on the theorems.     

A new weighted metric: the relative metric II

In the first part of this investigation we generalized a weighted distance function of R.-C. Li's and found necessary and sufficient conditions for it being a metric. In this paper some properties of this so-called M-relative metric are established. Specifically, isometries and quasiconvexity results are derived. We also illustrate connections between our approach and generalizations of the hyperbolic metric.     

Normal families of holomorphic functions and mappings on a Banach space

The authors lay the foundations for the study of normal families of holomorphic functions and mappings on an infinite-dimensional normed linear space. Characterizations of normal families, in terms of value distribution, spherical derivatives, and other geometric properties are derived. Montel-type theorems are established. A number of different topologies on spaces of holomorphic mappings are considered. Theorems about normal families are formulated and proved in the language of these various topologies. Normal functions are also introduced. Characterizations in terms of automorphisms and also in terms of invariant derivatives are presented.     

全纯函数族涉及公共值的正规定则

研究涉及公共值的全纯函数正规族问题．设F为单位圆△内的全纯函数族,a,b为两个判别的有穷复数且b≠0．文中证明了若对每一个,f∈F,Ef(a)=Ef(a),E↑-f′（b)包含于E^-f(b),则F在单位圆△内正规．     

全纯函数的分担值与正规族

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.     

Schwarz-Pick Inequalities for Hyperbolic Domains in the Extended Plane

Let and be two hyperbolic simply connected domains in the extended complex plane C¯ = C¯ {}. We derive sharp upper bounds for the modulus of the nth derivative of a holomorphic, resp. meromorphic function f: at a point z ₀ . The bounds depend on the densities (z ₀) and (f(z ₀)) of the Poincaré metrics and on the hyperbolic distances of the points z ₀ and f(z ₀) to the point .     

Shared values and normality

This paper investigates the relationship between the normality and the shared values for a meromorphic function on the unit disc Δ. Based on Marty’s normality criterion and through a detailed analysis of the meromorphic functions, it is shown that if for every f ∈ , f and f ^(k) share a and b on Δ and the zeros of f(z) − a are of multiplicity k ⩾ 3, then is normal on Δ, where is a family of meromorphic functions on the unit disc Δ, and a and b are distinct values. Selected from Journal of East China Normal University (Natural Science), 2003, 4: 12–18. This work was supported by the National Natural Science Foundation of China under grant number 10271122 and by Shanghai City Foundation for selected academic research     

拟正规与Hayman 选择

本文证明了亚纯函数族的一个拟正规定则, 推广了Xu-Fang, Chang 与Nevo-Pang-Zalcman 的相应结果; 并应用此拟正规定则得到了一个值分布结果, 推广与改进了著名的Hayman 选择, 以及Wang-Fang, Chang 和Nevo-Pang-Zalcman 的相关结论.     

关于亚纯函数的正规族

在本文研究了亚纯函数族的正规族问题,利用Zalcman引理的方法,获得了亚纯函数族的几个正规定则.并改进了顾永兴和Bergweiler的结果.     

关于亚纯函数的正规准则

在本文研究了亚纯函数族的正规族的问题,利用Zalcman引理的方法,获得了涉及亚纯函数族k阶导数的例外函数的几个正规准则.并改进了顾永兴[3]等人的结果.     

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G中正规。     

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.     

涉及分担值的亚纯函数族的几个正规定则

设(ぁ)为区域D上的一族亚纯函数,a,b为互相判虽的两个复数.若对(ぁ)中任意函数f,f在D内的极点重数至少为2,且当f(z)=a时,f'(z)=a;f(z)=b时f'(z)=b,则(ぁ)在D内正规.     

与例外函数和分担函数相关的亚纯函数的正规族

设F为区域D内的一族亚纯函数,对于每个f∈F,f的所有极点重数至少是2,a(z)和b(z)为两个在D内满足a(z)■b(z)的全纯函数.若对于每个f∈F,f(z)≠a(z)和f(z)≠b(z),则F在D内正规.这个结果改进了经典的Montel定则.此外,我们也讨论了亚纯函数族中每个函数与其导函数分担两个全纯函数的正规性.     

分担值和正规族

设F是单位圆盘△上的亚纯函数族,a是一个非零的有穷复数,如果f∈F,满足1)f的零点是重级的; 2)f和f'IM分担a, 则F在△上正规.相应于正规函数的结果也得到了证明.