Some Problems On Meromorphic Functions Concerning Sharing-value

In1920s, R. Nevanlinna introduced the characteristic functions of meromorphic functions and established the famous Nevanlinna theory, which had been significant breakthroughs in the development of the value distribution theory of meromorphic functions. Since then, Nevanlinna theory has been well developed itself and widely applied to the researches of normal family, uniqueness of meromorphic functions, complex differential equations and complex dynamics etc.Especially, With the de-velopment of cross subjects and the birth of new mathematical tools, many classis complex analysis problems have been solved, for instance, Problem on the Nevan-linna second fundamental theorem concerning small functions, Gross problem, Hay-man problem and so on. However, there are lots of new open questions to be posed, which are interested by many complex experts. In this thesis, we will study two main problems under the guidance of my supervisor, one is the value distribution and uniqueness of difference polynomial of meromorphic functions, the other is the uniqueness of meromorphic functions and their differential polynomials.The dissertation is structured as follows.In section one, we mainly introduce the basic Nevanlinna theory and some classical theory of the uniqueness results.In section two, we are mainly devoted to the value distribution of difference polynomials and obtain their related uniqueness theorems.We also study the unique-ness of meromorphic functions and their shifts concerning sharing-value and sharing-sets. Our theorems improve the results of I. Laine and C. C. Yang [Proc. Japan Acad. Ser. A,83(2007),148-151], K. Liu and L. Z. Yang [Arch. Math.92(2009),270-278], and J. Heittokangas, R. Korhonen, etc.[J. Math. Anal. Appl.355(2009),352-363]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.In section three, we investigate the uniqueness problem of meromorphic func-tions whose differential polynomials share one value CM. We improve some for-mer results given by Fang and Hua [49, Journal of Nanjing University Mathematical Biquarterly,13(1996),No.1,44-48], Yang and Hua [36, Ann. Acad. Sci. Fenn. Math.22(1997),No.2,395-406], Zhang and Lin[52, J. Math.Anal.Appl.,343(2008):938-950] and so on.In section four, we are mainly presented some problems.