Research On The Normal Criterion Of Meromorphic Functions

Normal families of meromorphic functions plays an important role in the complex analysis,and finding a new normal criterion is an important subject of normal families.In this thesis,we study the normality of meromorphic functions.This thesis consists of five chapters.The first chapter is introduction of the thesis.In the second chapter,we introduce some basic knowledge of Nevanlinna value distribution theory and the theory of normal families.In the third chapter,we studythe normal criterion of meromorphic functions with multiple zeros,and mainly prove the following theorem.Let M,n,k be three positive integers satisfying M?9 when n=k=1 and M???? when nk?1,b??0? be a finite complex number,and F be a family of holomorphic functions in a domain D.If for each f?F,?1?H?f?satisfying ??? is a differential polynomial of f;?2?all zeros of f have multiplicity at least k;?3?all zeros of fⁿf^?k??H?f?-b have multiplicity ?M,then F is normal in D.In the fourth chapter,we studythe normal criterion of meromorphic functions with shared values,and prove the following theorem.let F be a family of meromorphic functions in a domain D,k be a positive integer,a,b,c,d be four finite complexvalues,satisfying b ?a,0 and c ?0,d ?0.If for each f ?F,?1?all zeros of f d-have multiplicity atleast k;?2?f?z?=0?D?f?=a;?3?D?f?=b?f?z?=c;?4?k ?2 or ?a-b₀?z??^1/m??n?1??b-b₀?z?? when k=1,then F is normal in D.In the fifth chapter,we make asummary and outlook.The first result generalizes a normal criterion of Fang and Xu in 1993 involving the exceptional value,and the secondresultgeneralizes a normal criterion of Fang and Zalcmaninvolving the shared value s in the year of 2003.