On Shared Values And Normal Families Of Meromorphic Functions

In the beginning of last century ,Montel introduced the normal conception . In normality theory,finding new normality theorems is very important. In this paper we study normal families of meromorphic functions,which is an important subject in complex analysis.Much work has been made on this respect.The present paper is divided into four parts.In Chapter 1, we introduce some fundamental results,some definitions and some notations of Nevanlinna theory.In chapter 2, we obtained some results on normalities of holomorphic function families and its derivatives dealing shared polynomial CM,and mainly prove Theorem 2.1.2.Theorem 2.1.2 Let F be a family of holomorphic functions in a domain D,k(≥2) be a positive integer, let K be a positive number . Let P(z) be a polynomial . If for each f∈F,when f(z) = P(z), z∈D,have |f^(k)(z)|≤K,where f and f' shared P(z),then F is normal in D.In chapter 3, we obtained some results on normalities of holomorphic function families and its derivatives dealing shared entire function CM,and mainly prove Theorem 3.1.2 .Theorem 3.1.2 Let F be a family of holomorphic functions in a domain D ,k(≥2) be a positive integer and let a(z) be an entire function, K be a positive number, when f(z) =a(z),z∈D,have |f^(k)(z)|≤K,where f and f' shared a(z),then F is normal in D.In chapter 4, we obtained a result on normalities of meromorphic function families and f^(k) dealing shared a holomorphic function on the unitΔ,and mainly prove Theorem 4.1.1.Theorem 4.1.1 Let F be a family of meromorphic functions on the discΔ.Let a(z) be a non-zero holomorphic function and A; be a positive integer. If for everyf∈F,(1)f and f^(k) share a(z);(2)the zeros of f(z) are of multiplicity≥k+1 then F is normal onΔ.