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.     

关于亚纯函数分担值集正规族的注记(英文)

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.     

关于分担多项式的全纯函数的正规族(英文)

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.     

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.     

亚纯函数的一个正规定则

We proved:Let F be a family of meromorphic functions in a domain D and a≠0,b∈C.If f′(z)-a(f(z))~2≠b,f≠0 and the poles of f(z)are of multiplicity＞=3 for each f(z)∈F,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~(κ)-α are of multiplicity at least 2,and for each pair of functions f, g ∈ F, f~(κ)and g~(κ) share b in D, then F is normal in D.     

分担一个亚纯函数的亚纯函数族的正规定则

Let F be a family of meromorphic functions in D,and let Ψ(≠0) be a meromorphic function in D all of whose poles are simple.Suppose that,for each f ∈F,f≠0 in D.If for each pair of functions {f,g}(?) F,f' and g' share Ψ in D,then F is normal in D.     

A Normal Criterion Concerning Omitted Holomorphic Function

In this paper, we continue to discuss the normality concerning omitted holomorphic function and get the following result. Let F be a family of meromorphic functions on a domain D, k ≥ 4 be a positive integer, and let a(z) and b(z) be two holomorphic functions on D, where a(z) 0 and f(z) ≠ ∞ whenever a(z)=0. If for any f ∈ F, f'(z) -a(z)f^k(z) ≠ b(z), then F is normal on D.     

SOME NORMALITY CRITERIA OF MEROMORPHIC FUNCTIONS

Let F be a family of functions meromorphic in a domain D, let n ≥ 2 be a positive integer, and let a ≠ 0, b be two finite complex numbers. If, for each f ∈ F, all of whose zeros have multiplicity at least k ＋ 1, and f ＋ a（f^（k））^n≠b in D, then F is normal in D.     

ON THE NORMAL FAMILY OF MEROMORPHIC FUNCTIONS

In [1],W.K.Hayman has posed the following interesting problems: (A).Let be a family of holomorphic functions in the domain D.Let p be apositive integer,p≥3; and let a,b be two finite complex numbers,a≠0. If forany function f(z)∈ satisfies the condition f’(z)-a{f(z)}~P≠b,     

初等与非初等函数的关系

首先对现行教材中初等函数的定义提出了商讨意见,讨论了高等数学教材中出现的形式上的非初等函数与初等函数的关系,并通过一些有代表性的例子加以说明.     

CERTAIN SUFFICIENT CONDITIONS FOR STARLIKENESS AND CONVEXITY OF MEROMORPHICALLY MULTIVALENT FUNCTIONS

In this paper we derive certain suffcient conditions for starlikeness and con-vexity of orderαof meromorphically multivalent functions in the punctured unit disk.     

Certain Sufficient Conditions for Starlikeness and Convexity of Meromorphically Multivalent Functions

In this paper we derive certain sufficient conditions for starlikeness and convexity of order α of meromorphically multivalent functions in the punctured unit disk.     

TWO ENTIRE FUNCTIONS SHARING FOUR SMALL FUNCTIONS IN THE SENSE OF _(k))(β,f)=_(k))(β,g)

This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).     

奇变元Semibent-negabent函数的构造

研究negabent函数之间的关系,并利用bent-negabent函数构造出代数次数达到次最优的semibent-negabent函数.     

ALGEBROIDAL FUNCTION AND ITS DERIVED FUNCTION IN UNIT CIRCULAR DISC

In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.     

关于亚纯函数导数的特征函数的一点注记

In this paper,the characteristic function of the derivative of meromorphic func- tion is studied.A expression of characteristic function T(r,f~1)is given.     

TWO ENTIRE FUNCTIONS SHARING FOUR SMALL FUNCTIONS IN THE SENSE OF (E-)k)(β, f)= (E-)k)(β,g)

This paper proves a result that if two entire functions f(z) and g(z) share foursmall functions aj (z) (j = 1,2, 3, 4) in the sense of Ek) (aj, f) = Ek)(aj, g), (j = 1, 2, 3, 4)(k ≥ 11), then there exists f(z) = g(z).     

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.