Several Normality Criterion In Relation To Meromorphic Functions

In 1920s,R.Nevanlinna introduced the characteristics functions of meromorphic functions and gave the famous Nevanlinna theory.This theory is one of the most important achievements in twenty centuries.For over a half century,Nevanlinna theory has been well developed and can be used in the research of meromorphic functions. Especially,there have been a lot of celebrated results in the normal famlity theory.In this thesis,by using Nevanlinna theory,we gave some researches on the normal famility theory of meromorphic function under the guidance of Mr. Li jiangtao.There are three sections in the thesis.In section 1,we introduce some fundamental results, some notations of Nevanlinna theory.In section 2,we study the normalty criterion of meromorphic functions.A general normality criteria is abtained,which improves the results given by Wang yuefei and Fang Mingliang,Zhang Wenhua,Li Jiangtao and Yi Hongxun.That is f (k) is replaced by L(f),where L(f)= f^k(z)+a_k-1 (z)f^k-1(z)++a₀(z)f(z), a_k-1 ( z),,a₀(z) be some holomorphic functions in a domain D .In section 3,we study the normal families of holomorphic functions concerning sharing polynomial,therefore we improved the results of Xu Yan. In this thesis,we obtain the follow results:Theorem 1 Let F be a family of meromorphic functions in a domain D , n, kbe a positive integer with n≥k+1,b a finite complex number,εbe a positive number,and let L(f)= f^k(z)+a_k-1 (z)f^k-1(z)++a₀(z)f(z),where a_k-1 ( z),,a₀(z) be some holomorphic functions in a domain D . If for each f∈F, the zeros of f have multiplicity at least n ,the poles of f have multiplicity at least 2,and L(f)= b?f( l )(z)≥ε(0≤lk(z)+a_k-1 (z)f^k-1(z)++a₀(z)f(z),where a_k-1 ( z),,a₀(z) be some holomorphic functions in a domain D .If for each f∈F,(1) f and L(f)shared valuea ;(2) the zeros of f have multiplicity at least k+1; then F is normal in D . Theorem 3 Let F be a family of holomorphic functions in a domain D . M, Nbe two positive numbers,and let L(f)= f( k )(z)+a_k-1 (z)f^k-1(z)++a₀(z)f(z),where a_k-1 ( z),,a₀(z) be some holomorphic functions in a domain D . If for each f∈F, the zeros of f have multiplicity at leastk , a ( z)≠0 be a holomorphic function,and L(f)= a(z)?f(z)≥N,f(z)=0?f( k )(z)≤M, then F is normal in D .Theorem 4 Let F be a family of holomorphic functions in a domain D . Letk be a positive integer, a , b≠0,c≠0be finite complex number,and let L(f)= f( k )(z)+a_k-1 (z)f^k-1(z)++a₀(z)f(z),where a_k-1 ( z),,a₀(z) be some holomorphic functions in a domain D . If for each f∈F, the zeros of f have multiplicity at leastk ,and f = 0 ?L(f)=a,L(f)=b?f=c, then F is normal in D .Theorem 5 Let F be a family holomorphic functions in a domain D , k (≥2) be a positive integer.If for each f∈F,fand f′share R (z),where R (z) be a polynomial.and f^k(z)= R(z)whenever f ( z)= R(z),z∈D, then F is normal in D .Theorem 6 Let F be a family of holomorphic functions in a domain D , k (≥2) be a positive integer, K be a positive number,and R (z) be a polynomial.If for each f∈F, f and f′share R (z),and f^k(z)≤Kwhenever f ( z)= R(z)in D ,then F is normal in D .