Some Problems On Normal Families And Shared Values Of Meromorphic Functions

Since P.Montel introduced the concept of normal families in the early 20th century,the normality of function families has always been an important research subject in the single complex function of complex analysis.It has a very wide application in the aspects of the uniqueness of meromorphic mapping and the existence of singular direction.Its development continues to this day,many scholars at home and abroad have made effective results after a considerable process.In particular,W.Schwick first considered combining the normality and uniqueness of function families in 1992 and since then the discussion about the normality and shared values of meromorphic functions has become one of the important parts in the theoretical system of normal families of meromorphic functions.Although the research results in this field are continuously improving and complete,the value and universality of its application still make the research in-depth.This paper starts with the work of scholars at home and abroad,takes Nevanlinna value distribution theory as the fundamental premise,and focuses on the use of Zalcman Pang lemma to continue to explore the meromorphic function families related to shared values and determine its normality.That is to say,some conclusions of normal families of meromorphic functions and shared values or shared sets or shared functions are generalized and improved,and corresponding new normal rules are obtained respectively,which further enriches the theoretical system of normal families and provides theoretical basis for its application.The paper is divided into five chapters,specifically: In Chapter 1,we mainly show the development of normal families theory and the purpose,method,process and results of this paper;In Chapter 2,we introduce some preparatory knowledge,display the basic notations and important theorems about Nevanlinna value distribution theory,as well as the essential theories of normal families and shared values of meromorphic functions;In Chapter 3,we talk about the main research findings and related research background of this paper,it is subdivided into normal criterions related to shared values,shared sets and shared functions,which generalizes the conclusions of Liu Zhihong,Han Minghua,Chang Jianming,etc.;In chapter 4,we fisrt list the lemmas needed to prove each theorem,and then discuss the proving process in detail;In chapter 5,we summarize the results of this paper and two questions that are expected to be solved.