Further Results On Normal Families And Exceptional Functions

At the beginning of the twentieth Century,Montel defined normal family:Let D be a domain in the complex plane C,and F be a family of meromorphic functions defined on D.F is said to be normal on D,if for any sequence {fn}(?)F there exists a subsequence {fnk},such that {fnk} converges spherically locally uniformly on D,to a meromorphic function or oo,then F is said to be normal on D(see[7],[12],[20]).Montel gave the Montel Normality Criterion:Let F be a family of meromorphic functions defined in D,a,b,c is three different complex numbers,if for any function f ? F,f(z)? a,b,c,then F is normal in D.Montel Normal Criterion first connects the normality of meromorphic families with the value of functions.Later,Gu-Miranda linked the normality and derivative:Let F be a family of meromorphic functions defined in D,k be a,positive integer,if for any function f ? F,f(z)? 0 and f(k(z)? 1,then F is normal in D.In recent years,many experts generalized the exceptional values to exceptional functions and get some meaningful results.This thesis is divided into two parts,which are arranged as follows:In Chapter 1,we introduce the Nevanlinna value distribution theory and the normal family the-ory;in Chapter 2,we generalized the normality criteria about exceptional functions of Pang Xuecheng,Xu Yan and others,got further results on normal families and exceptional functions,as follow:Theorem 2.1.1 Let k ? 4,?(z)((?)0)be a function holomorphic in a domain D,and F be a family of meromorphic functions defined in D.If,for every function f ? F,f has only zeros of multiplicity at least k and satisfies the following condi-tions:(a)f(z)= 0(?)|f(k)(z)| ? A|?(z)|.(b)f(k)(z)??(z).(c)All poles of f have multiplicity at least 2.Then F is normal in D.Theorem 2.1.2 Let ?(z)((?)0)be a function holomorphic in a domain D,and F be a family of meromorphic functions defined in D.If,for every function f ? F,f has only zeros of multiplicity at least 3 and satisfies the following conditions:(a)f(z)= 0(?)|f(3)(z)| ? A|?(z)|.(b)f(3)(z)= ?(z).(c)All poles of f have multiplicity at least 3.Then F is normal in D.Theorem 2.1.3 Let l be a positive integer,and ?(z)((?)0)be a function holomorphic in a domain D,all of whose zeros have multiplicity at most l.Let F be a family of meromorphic functions defined in D.If,for every function f ? F,f has only zeros of multiplicity at least 2 and satisfies the following conditions:(a)f(z)=0(?)|f"(z)|?A|?(z)|.(b)f"(z)??(z).(c)All poles of f have multiplicity at least l + 3.Then F is normal in D.