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 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 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 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.

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 of Holomorphic Functions Concerning Zero Numbers

In this paper, we study the normal families related with a Hayman conjecture of higher derivative concerning zero numbers, and get one normal criteria.Our result improve some earlier related result.

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 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.

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D C C, all of whose zeros have multiplicity at least k, where k 〉 2 is an integer. And let h（z）≠ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈F： （a） f（z） = 0 [f^（k）（z）｜ 〈 ｜h（z）｜; （b） f^（k）（z）≠ h（z）. Then F is normal on

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 C, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. Let h（z） ≠ 0 and oo be a meromorphic function on D. Assume that the following two conditions hold for every f C Dr ： （a） f（z） = 0 =→ ｜f（k）（z）｜ 〈｜h（z）｜. （b） f（k）（z） ≠ h（z）. Then F is normal on D.

Quasi-normal Family of Meromorphic Functions Whose Certain Type of Differential Polynomials Have No Zeros

Define the differential operators φ_(n) for n∈N inductively by φ_(1)[f](z)=f(z)and φ_(n+1)[f](z)=f(z)φ_(n)[f](z)+d/daφ_(n)[f](z).For a positive integer k≥2 and a positive number δ,let F be the family of functions f meromorphic on domain D■C such that φ_(k)[f](z)≠0 and|Res(f,a)-j|≥δ for all j∈{0,1,...,k-1}and all simple poles α of f in D.Then F is quasi-normal on D of order 1.

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.

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.

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.

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.

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 Functions Concerning Shared Values

In this paper we discuss normal functions concerning shared values. We obtain the follow result. Let F be a family of meromorphic functions in the unit disc A, and a be a nonzero finite complex number. If for any f ∈F, the zeros of f are of multiplicity, f and f′ share a, then there exists a positive number M such that for any f∈F1（1-｜z｜^2） ｜f′（z）｜/1＋｜f（z）｜^2≤M.