Normal Families and Shared Functions

Let k be a positive integer,let h be a holomorphic function in a domain D,h■0and let F be a family of nonvanishing meromorphic functions in D.If each pair of functions f and q in F,f^((k)) and g^((k)) share h in D,then F is normal in 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.

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 MEROMORPHIC FUNCTIONS WITH SHARED VALUES

We obtain some normality criteria of families of meromorphic functions sharing values related to Hayman conjecture, which improves some earlier related results.

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.

A Note on Normal Families of Meromorphic Functions and Sharing Set

In the paper,we prove the main result:Let k(≥2)be an integer,and a,b and c be three distinct complex numbers.Let F be a family of functions holomorphic in a domain D in complex plane,all of whose zeros have multiplicity at least k.Suppose that for each f∈F,f(z)and f(k)(z)share the set{a,b,c}.Then F is a normal family in D.

Normal families and shared values of meromorphic functions

We studied the normality conditions in families of meromorphic functions, improved the results of Fang and Zalcman [Fang ML, Zalcman L, Normal families and shared values of meromorphic functions, Computational Methods and Function Theory, 2001, 1 （1）： 289-299], and generalized two new normality criterions. Let F be a family of meromorphic functions in a domain D, a a non-zero finite complex number, B a positive real number, and k and m two positive integers satisfying m〉2k＋4. If every function denoted by f belonging to F has only zeros with multiplicity at least k and satisfies f^m（z）f^（k）（Z）=α→ ｜^f（k）（z）｜ ≤B or f^m（z）f^（k）（z）=α→｜f（z）｜ ≥, then F is normal in D.

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^（cz）/u ＋（ac-a^2）/c.The proof uses the theory of normal families in an essential way.

Normal families related to shared sets

We studied the normality criterion for families of meromorphic functions related to shared sets. Let F be a family of meromorphic functions on the unit disc △, a and b be distinct non-zero values, S={a,b}, and k be a positive integer. If for every f∈ F, i） the zeros of f（z） have a multiplicity of at least k＋ 1, and ii） E^-f（k）（S） lohtain in E^-f（S）, then F is normal on .4. At the same time, the corresponding results of normal function are also proved.

Normal families of meromorphic functions

A normal theorem concerning meromorphic functions sharing values was proved with the method of Zalcman- Pang.The theorem is as follows. If for each f in F, all zeros of f-a have multiplicity at least k (k≥2), f and its k-th derivative function share a, and if f=b whenever its k-th derivative equal b, then F is normal in D. This theorem improved the result of Chen and Fang [Chen HH, Fang ML, Shared values and normal families of meromorphic functions, Journal of Mathematical Analysis and Applications, 2001, 260: 124-132].

Normal Families of Meromorphic Functions and Shared Set

In this paper,we study the normality criterion for families of meromorphic functions concerning shared set depending on f∈F.Let F be a family of meromorphic functions in the unit disc A.For each f∈F,all zeros of f have multiplicity at least 2 and there exist nonzero complex numbers b_f,c_f satisfying(i) b_f/c_f is a constant;(ii) min{σ(0,b_f),σ(0,c_f),σ(b_f,c_f)} ≥m for some m > 0;(iii) E_f'(S_f)■ E_f(S_f),where S_f = {b_f,c_f}.Then F is normal in A.At the same time,the corresponding results are also proved.The results in this paper improve and generalize the related results

Shared Hyperplanes and Normal Families of Holomorphic Curves

In theorem LP [1], Liu proves the theorem when N = 2, but it can’t be ex-tended to the general case in his proof. So we consider the condition that the families of holomorphic curves share eleven hyperplanes, and we get the theorem 1.1.

NORMALITY AND THE NUMBER OF ZEROS

In this paper,we study normal families of meromorphic functions.By using the idea in[11],we obtain some normality criteria for families of meromorphic functions that concern the number of zeros of the differential polynomial,which extends the related result of Li,and Chen et al..An example is given to show that the hypothesis on the zeros of a(z)is necessary.

Normal Families and Uniqueness Related to Shared Values

Let F be a family of meromorphic functions on the unit disc A. Let a be a non-zero finite value and k be a positive integer. If for every f ∈ F,(i) f and f(k) share α ;(ii) the zeros of f(z) are of multiplicity ≥k + 1 , then F is normal on △.We also proved corresponding results on normal functions and a uniqueness theorem of entire functions .

NORMAL CRITERION FOR FAMILIES OF HOLOMORPHIC MAPS OF SEVERAL COMPLEX VARIABLES INTO P^N(C)WITH MOVING HYPERSURFACES

This article gives a normal criterion for families of holomorphic mappings of several complex variables into P N(C)for moving hypersurfaces in pointwise general position,related to an Eremenko’s theorem.

Differential Polynomials and Normal Families

This paper continues the researches of Yang Lo and others, and gives a further generalization of Gu’s criterion for the normality of a family of meromorphic functions.

Picard Values and Normal Families of Meromorphic Functions with Multiple Zeros

In this paper, general modular theorems are obtained for meromorphic functions and their derivatives. The related criteria for normality of families of meromorphic functions are proved.

Normal Families and Uniqueness of Entire Functions and Their Derivatives

Let f be a nonconstant entire function; let k ≥ 2 be a positive integer; and let a be a nonzero complex number. If f（z） = a→f′（z） = a, and f′（z） = a →f^（k）（z） = a, then either f = Ce^λz ＋ a or f = Ce^λz ＋ a（λ - 1）/）λ, where C and ）, are nonzero constants with λ^k-1 = 1. The proof is based on the Wiman-Vlairon theory and the theory of normal families in an essential way.

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.

Radically distributed value and normal families of meromorphic functions

We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.