The Normal Family Of Meromorphic Functions Related To Zero Numbers And Shared Set

The theory of normal families of meromorpfic is an important part in complex analysis,many mathematical workers who are at home and abroad have made outstanding contributions and attained lots of research results.In this thesis,two results are proved.The first one is a normality criterion concerning zero numbers and polynomial:Let F be a family of zero-free meromorphic functions in a domain D:let h(z)be a holomophic function in D.and let k,q be two positive integers.If,for each function f?F,f(z)?0,(f(k)(z))q-(h(z))q have at most q(k + 1)-1 distinct zeros(ignoring multiplicity)in D,then F is normal in D.The other is a normality criterion concerning shared set:Let F be a family of meromorphic functions in a domain D,let a,b,c be finite complex numbers,and a ?b,let k,q(?1)be two positive integers and S = ?a,b?.If,for each function f?F,the zeros of f-c are of multiplicity?k+1,and f?S whenever(f(k)(z))q?S,then F is normal in D.This thesis consist of six chapters,The first chapter mainly introduces the research status on normal families of meromorphic functions.The second and third chapters introduce some definitions and theorems about the value distribu-tion theorem and the normal families.In the fourth and fifth chapters,we give our results,proofs and so on.In the last chapter,we give some open questions.