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 FAMILY OF MEROMORPHIC FUNCTIONS SHARING HOLOMORPHIC FUNCTIONS AND THE CONVERSE OF THE BLOCH PRINCIPLE

In 1996, C. C. Yang and P. C. Hu [8] showed that： Let f be a transcendental meromorphic function on the complex plane, and a ≠ 0 be a complex number; then assume that n 〉 2, n1,… , nk are nonnegative integers such that n1＋… ＋ nk ≥1; thus fn（f′）n1…（f（k））nk-a has infinitely zeros. The aim of this article is to study the value distribution of differential polynomial, which is an extension of the result of Yang and Hu for small function and all zeros of f having multiplicity at least k ≥2. Namely, we prove that fn（f′）n1…（f（k））nk-a（z） has infinitely zeros, where f is a transcendental meromorphic function on the complex plane whose all zeros have multiplicity at least k≥ 2, and a（z） 0 is a small function of f and n ≥ 2, n1,… ,nk are nonnegative integers satisfying n1＋ …＋ nk ≥1. Using it, we establish some normality criterias for a family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. The results of this article are supplement of some problems studied by d. Yunbo and G. Zongsheng [6], and extension of some problems studied X. Wu and Y. Xu [10]. The main result of this article also leads to a counterexample to the converse of Bloeh＇s principle.

NORMAL FAMILY OF COMPOSITIONS OF HOLOMORPHIC FUNCTIONS AND THEIR HIGH ORDER DERIVATIVES

Let F = {f} be a family of entire functions and f(k) the k-th derivative of f. The author studies the relation of the nomality of F and that of {f(k) o f : f ∈F} or {f o f(k):f∈F}.

The Normal Family of Meromorphic Functions Concerning Shared Analytic Function

In this paper,we study the normal criterion of meromorphic functions concerning shared analytic function.We get some theorems concerning shared analytic function,which improves some earlier related results.

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.

ON ENTIRE FUNCTIONS, BLOCH AND NORMAL FUNCTIONS

New criteria for an analytic function to be Bloch and for a meromorphic function to benormal are given. These criteria generalize the recently introduced area integral conditionsinvolving a Green's function.

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 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 family and the sequence of omitted functions

Let {fn} be a sequence of meromorphic functions on a plane domain D, whose zeros and poles have multiplicity at least 3. Let {hn} be a sequence of meromorphic functions on D, whose poles are multiple, such that {hn} converges locally uniformly in the spherical metric to a function h which is meromorphic and zero-free on D.If fn≠hn, then {fn} is normal on D.

Normal family of algebroidal functions

The multivalue algebroidal functions are studied with geometric method. The difficulties of multivalue and branch points are overcome, and some theorems on normality are obtained.

THE NORMAL FAMILITY OF MERMORPHIC FUNCTIONS

Let F be a family of mermorphic functions in a domain D, and let a, b, c be complex numbers, a ≠ b. If for each f ∈ F, the zeros of f-c are of multiplicity ≥ k ＋ 1, and -↑Ef（k）（a） belong to -↑Ef （a）, -↑Ef（k）（b）belong to -↑Ef （b）, then F is normal in D.

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 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 Normality Criteria for Families of Meromorphic Functions

Let κ be a positive integer and F be a family of meromorphic functions in a domain D such that for each f ∈ F, all poles of f are of multiplicity at least 2,and all zeros of f are of multiplicity at least κ + 1. Let α and b be two distinct finite complex numbers. If for each f ∈ F, all zeros of f;-α are of multiplicity at least 2,and for each pair of functions f, g ∈ F, f;and g;share b in D, then F is normal in D.

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

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].

Normality Criteria of Meromorphic Functions Concerning Shared Analytic Function

In this paper, we use Pang-Zalcman lemma to investigate the normal family of meromorphic functions concerning shared analytic function, which improves some earlier related results.