Periodicity And Uniqueness Of Meromorphic Functions With Shared-values

The uniqueness theory of meromorphic functions bases on the value distribu-tion theory of meromorphic functions established by R. Nevanlinna. That is, we can uniquely determine a meromorphic function by assuming some values, such as Nevanlinna's five values theorem and four values theorem. In this region, many mathematicians such as H.X.Yi, C.C.Yang et.al had done a lot of important works. Today, the value distribution theory and the uniqueness theory of meromorphic functions are still an importance branch of complex function theory.Recently, Halburd and Korhonen studied the difference analogues of Nevan-linna's theory. Later, Heittokangas considered the uniqueness problem of meromor-phic function f(z) with its shift f(z+c). In this paper, based on these fundamental results, we firstly study the periodicity problems and uniqueness problems of mero-morphic functions with three shared-values. And then, we study the uniqueness problems of meromorphic functions with its shift considering two shared-sets. Ad-dition, we study the uniqueness problem of meromorphic functions sharing two sets. Also, the whole paper is divided into four chapters.In chapter 1, we briefly introduce the Nevanlinna's theory and the difference analogues of Nevanlinna's theory.In Chapter 2, we deal with the uniqueness and periodicity of meromorphic functions of finite order with two shared-values CM and one IM (2CM+1IM), which respectively improves one result of Brosch's and one result of J.H.Zhen with three shared-values CM (3CM). Moreover, examples shows that the condition is sufficient and sharp possibly.In Chapter 3, we considered the two shared-set problems of meromorphic func-tion f(z) with their shifts f(z+c), which extend some results of J.L.Z and so on.In Chapter 4, we are concerned with the uniqueness problems of meromorphic functions with two shared-sets. which improve some results of H.X.Yi, J.T.Li and so on.