Normality Criterion Of Families Of Meromorphic Functions With Derivative

The concept of normal family was founded by P.Montel in the last century and promoted by the value distribution theory of R.Nevanlinna.It plays an important role in the complex dynamic system and the Julia set.In this paper,some normality rules for meromorphic functions are studied on the basis of the derivatives,and the following results are obtained.This article is divided into four chapters:The first chapter is the introduction,which introduces the research background of the normal family,the current research situation at domestic and abroad,and some necessary preparatory knowledge and conclusions.In the second chapter,a new criterion of normal family is proved,under the following conditions:let(?)be a family of meromorphic functions in a domain D,k and l(≥2)be positive integers,and b a non-zero complex number.The criterion that if for each f(z)∈(?),the zeros of f(z)are of multiplicity at least k +l,and f(k)(z)=0(?)f(k+l)(z)=0,f((k+l)(z)=b(?)f(k)(z)=b,then(?)is normal in D,is established on the base of one of results we obtained,and is specially for the case about the derivative function with kth-order and(k + l)th-order;In the third chapter,some new criteria of normal family are proved,and one of them further states that any family(?)of meromorphic functions in a domain D,whose every member’s zeros are of multiplicity at least k and the member’s kth-order derivative takes values at the zeros with a uniform finite boundary,will be normal if the family consisted of the kth-order derivatives of all member’s of(?)is normal.As shown in the paper,the result is specially for the case concerning derivatives of functions with kth-order;In the last chapter,one criterion of normal family is given on the basis of the third chapter,it shows that if there exist a constant C,an integer number k>2,and a compact set E in a domain D such that for each f(z)of a normal family(?)of meromorphic functions in D,the derivative f’(z)and the kth-order derivative f(k)(z)have no zeros,and |f(k)(z)|>C|f(z)| holds for some points in E,then the familyconsisted of derivatives of all members of(?)is also normal.