Normal Family And Uniqueness Of Meromorphic Functions Involving Differential Polynomials

In 1925,R.Nevanlinna introduced the characteristic function of meromorphic functions and gave two basic theorems,thus establishing the Nevanlinna value distribution theory of meromorphic functions,which provided the research tools and theoretical basis for the theoretical research on the normal family and uniqueness of meromorphic functions.The theory of normal family of meromorphic functions is an important part of complex analysis.At the beginning of last century,P.Montel introduced the concept of normal family.He called the family of functions with some list compactness normal family.Then came the famous Marty theorem and Miranda,Valiron and Zhuang Qitai’s normality.In 1975,L.Zalcman introduced the necessary and sufficient conditions of the not normal meromorphic function family,which made the development of the normal family theory become mature.The uniqueness theory of meromorphic functions is an important research topic in complex analysis.The uniqueness theory of meromorphic functions,generally speaking,is that a function can be uniquely determined by what conditions,or two functions are identical when they satisfy what conditions.In 1929,R.Nevanlinna proved the famous five-value uniqueness theorem,and then he obtained the famous four-value uniqueness theorem.These are two classical results of the uniqueness theory of meromorphic functions.For decades,the research on the uniqueness theory of meromorphic functions has been very active.Domestic and foreign complex analysis scholars include:E.Mues,G.Frank,M.Reinders,N.Steinmetz,G.G.Gundersen,C.C.Yang,H.X.Yi and others have obtained many excellent results in the study of the uniqueness of meromorphic functions.The main work of my doctoral thesis is to discuss the normality and uniqueness of meromorphic functions involving differential polynomials.The thesis is divided into some parts.The first chapter introduces the research background,content and programmes of this thesis.The second chapter briefly introduces the Nevanlinna theory,and gives some definitions and theorems used in this thesis.In chapter 3,we study the normal family of meromorphic functions involving differential polynomials in a kind of Hayman’s problems.We generalize the result of Gu Yongxing and the result of Fang Mingliang et al.,from the two cases of meromorphic functions without zero and meromorphic functions with zero multiplicity not less than k+1.In chapter 4,we study the meromorphic solutions of a certain type of nonlinear differential equations,and improve the results of Chen Ming-feng and Gao Zong-sheng by using the method of classification discussion,that is,the conditions of"transcendental function" and " finite order" are removed.The fifth chapter puts forward some problems to be solved for further study.