Studies On Some Problems About The Uniqueness Of Meromorphic Functions

In the early 20th century,Nevanlinna established the Nevanlinna value distribution theory,which has important applications in many aspects,such as the uniqueness theory of meromorphic function,the theory of normal family,the theory of complex differential equation and complex dynamical system.In this paper,some problems about the uniqueness of meromorphic functions are studied by using Nevanlinna value distribution theory.This paper is mainly divided into six parts:In the first part,we introduce the research background of Nevanlinna value distribution theory,uniqueness of meromorphic functions,L-functions of Selberg class and periodicity of entire functions.In the second part,we introduce the value distribution theory of meromorphic function,the difference anologue theory,the L-function theory of Selberg class,the uniqueness theory of meromorphic function and some basic concepts and notations.In the third part,we mainly study the uniqueness of two meromorphic functions,f and g,under the condition of reduced sharing value,that is,when[f(z)]n and[g(z)]n share a CM,f(z)and g(z)share ? IM,and applies the results to the difference.In the fourth part,we investigate the uniqueness problems of a differential polynomials[fn(f-1)m](k)sharing a small function with L-function[Ln(L1)m](k)by using the notion of weakly weighted sharing and relaxed weighted sharing.In the fifth part,we mainly study the periodicity of entire function.Around the conjecture of Yang C C,when f has a Picard exceptional value,if f'(z)-afn(z)is a periodic function,we prove that f is also a periodic function.In the sixth part,we summarize the work done in this paper and look forward to the future work.