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.

On Normal Criterion of Meromorphic Functions

We proved： Let F be a family of meromorphic functions in a domain D anda α ≠0, b ∈C. If f1（z） - α（f（z））^2 ≠ b, f≠ 0 and the poles of f（z） are of multiplicity ≥ 3 foreach f（z） ∈F, then F is normal in D.

Normality Criteria of Meromorphic Functions Concerning Shared Fixed-points

In this paper, we study the normality criteria of meromorphic functions concerning shared fixed-points, we obtain: Let F be a family of meromorphic functions defined in a domain D. Let n, k ≥ 2 be two positive integers. For every f ∈ F, all of whose zeros have multiplicity at least (nk+2)/(n-1). If f(f(k))nand g(g(k))nshare z in D for each pair of functions f and g, then F is normal.

The Value Distribution and Normality Criteria of a Class of Meromorphic Functions

In this article,we use Zalcman Lemma to investigate the normal family of meromorphic functions concerning shared values,which improves some earlier related results.

Normality Criteria of Meromorphic Functions

In this paper, we consider normality criteria for a family of meromorphic functions concerning shared values. Let F be a family of meromorphic functions defined in a domain D, m, n, k and d be four positive integers satisfying m ≥ n ＋ 2 and d≥k＋1/m-n-1 and a（≠ 0）, b be two finite constants. Suppose that every f∈F has all its zeros and poles of multiplicity at least k and d, respectively. If （fn）（k）-afm and （gn）（k） -agm share the value b for every pair of functions （f, g） of ~, then is normal in D. Our results improve the related theorems of Schwick （Schwick W. Normality criteria for families of meromorphic function. J. Anal. Math., 1989, 52： 241-289）, Li and Gu （Li Y T, Gu Y X. On normal families of meromorphic functions. J. Math. Anal. Appl., 2009, 354： 421-425）.

Some Notes on Normality Criteria of Meromorphic Functions

In this paper, we study the normality of families of meromorohic functions related to a Hayman conjecture. We prove that the conditions in Hayman conjecture and in other criterions can be relaxed. The results in this paper improve some previous results.

The Normal Meromorphic Functions Family Concerning Higher Order Derivative and Shared Set by One-Way

Let F be a meromorphic functions family on the unit disc Δ, If for every (the zeros of f is a multiplicity of at least k) and if then and ( ), then F is normal on Δ.

Normality of Meromorphic Functions Family and Shared Set by One-way

We studied the normality criterion for families of meromorphic functions which related to One-way sharing set, and obtain two normal criterions, which improve the previous results.

涉及微分多项式的亚纯函数正规定则

研究关于微分多项式的亚纯函数的正规定则,利用了Pang-Zalcman引理,将危合文和张惠玲的一些结果进行推广,得到了关于微分多项式以及分担值的亚纯函数的正规定则的结果。设F是定义在区域D上的一个亚纯函数族,若对于任一的f∈F, f的零点重级≥k+1,并且满足f+a(L(f))^(n)≠b(z), n≥k+2,其中k,n均为正整数,a≠0为常数,b(z)≠0在区域D内解析,则F在区域D内正规。设F是定义在区域D内的一族亚纯函数,族中任一函数的零点重级≥k+1, a,b均为非零有穷复数。任一取f, g∈F,若f+a(L(f))^(n),g+a(L(g))^(n)分担b, n,k∈N*,并且n≥2,则F在区域D内正规。

BLOCH'S PRINCIPLE AND NORMAL CRITERION

In this paper, Bloch’s principle is discussed and a normal criterion is asserted. If Fis a family of meromorphic functions in domain D, for every function f∈F then F is normal in D.

Normality Family and Shared Functions by Meromorphic Functions and Its Differential Polynomials

In this paper, we obtain the following normal criterion： Let F be a family of mero-morphic functions in domain D belong to C, all of whose zeros have multiplicity k ＋ 1 at least. If there exist holomorphic functions α（z） not vanishing on D, such that for every function f（z） ∈F, f（z） shares α（z） IM with L（f） on D, then F is normal on D, where L（f） is linear differential polynomials of f（z） with holomorphic coefficients, and k is some positive numbers. We also proved coressponding results on normal functions.

亚纯函数的正规定则

文章通过幅角原理和Zalcman-Pang引理,利用构造辅助函数的方法,对满足条件f′-a(z)f k≠b(z)的亚纯函数族的正规性进行研究。首先,讨论当k=3,例外函数a(z),b(z)均为全纯函数时,在函数族零点重级的条件下,亚纯函数族是正规的,完善了正规族已有结论;其次,将例外函数由全纯函数推广为亚纯函数,讨论例外函数a(z),b(z)分别为亚纯函数时的正规性,得到当a(z)是亚纯函数时,全纯函数族是正规的,当b(z)是亚纯函数时,无零点的亚纯函数族是正规的,在一定范围内推广了亚纯函数的现有结果。

涉及复合亚纯函数和不动点的亚纯函数的正规族（英文）

假设f是一个超越整函数,G是定义在区域D■C上的全纯函数族.如果对G中每个元素g,f(g)-α1在区域D上的每个零点重数≥2,f(g)-α2和g-α2在区域D上IM分担0,这里α1和α2是2个判别的有穷复数,则G在区域D上是正规的,该结果推广了Bergweiler 2004年的一个结果.同时还证明了:假设R是一个次数满足deg R≥2(deg R≥3,并且R在在复平面上有3个判别的有限的不动点)的有理函数,F是一个定义在区域D■C上的全纯函数(亚纯函数),并且对F中每个元素f,Rf(z)-z和f(z)-z在区域D上IM分担0,则F是区域D上的正规族,该结果推广了方明亮与袁文俊2000年的一个结果,也推广了常建明、方明亮与L.Zalcman 2005年的一个结果,并举例说明本文结果从某种意义上来讲是最佳的.

与分担值相关的亚纯函数的正规性

设F是区域D内的亚纯函数族,a,b,c,d,k,l是有穷复数,c≠k,b≠d,b≠0,l≠0,F中任意的函数f满足:(1°)若f′=b,则f=a;(2°)若f=c,则f′=d;(3°)若f=k,则f′=l,

关于亚纯函数族的两个正规定则

本文证明了两个正规定则：（ｉ）设Ｍ为区域Ｄ内的亚纯函数族，ａ０（ｚ），…，ａｋ－１（ｚ）为Ｄ内ｋ个全纯函数。若，ｆ的每个零点之级≥ｍ，而ｇ（ｚ）－ｂｊ（ｊ＝１，２；ｂ１，ｂ２∈Ｃ＼｛０｝，ｂ１≠ｂ２）的每个零点之级分别≥ｎｊ，这里ｇ（ｚ）…＋ａ０（ｚ）ｆ（ｚ），并且（ｋ＋１），则Ｍ在Ｄ内正规。（ｉｉ）设Ｆ为区域Ｄ内的全纯函数族，为给定的常系数多项式，ｎ≥１。若有ｆ＇Ｐ（ｆ）在Ｄ内不取１，则Ｆ在Ｄ内正规。

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) Ef(b),则F在单位圆△内正规.

全纯函数涉及公共值的正规定则

本文研究全纯函数涉及公共值的正规族问题．设F为单位圆Δ内的全纯函数族,a, b与c为3个有穷复数,且b≠0,c≠0,a≠b．若对每个f∈F,f的零点重级至少是k,并且E^-_f(0)=E^-_f(k)(a),E^-_f(k)(b)(?)E^-_f(c),则F在单位圆Δ内正规．

关于亚纯函数的正规定则

证明了如果一个亚纯函数f满足f(n)(z)-a(f(z))n+1≠b,其中,n∈N,n≥1,a(≠0)和b是两个判别的有穷复数,f的极点重级至少为n+2,f没有零点,或者零点的重级至少为n+3,则f是常数.同时也得到了相应的正规定则.

涉及微分多项式分担函数的正规定则

设m(≥0)是一个正整数,h(z)(≠0)是区域D内的全纯函数,且其零点重级均≤m,P是多项式满足deg P≥3,或者degP=2且P仅有一个零点.设F是区域D内的一族亚纯函数,其零点与极点重级均≥m+1.如果对于F中的任意两个函数f,g,P(f)f′与P(g)g′分担h(z),则F在区域D内正规.该结果改进了Lei and Fang^([8]),Zhang^([16])等人的结果.

涉及零点数的亚纯函数的正规族

该文研究了涉及零点个数的亚纯函数的正规族,得到了三个正规定则.这些结论推广了前人的结论.