Uniqueness Results Concerning Difference Polynomials Of Meromorphic Functions

In1920s, R.Nevanlinna, a famous Finnish mathematician, establishedNevanlinna theory. The theory is the important tool of theoretical research complexand is surely one of the most important achievements in mathematics in the20thcentury. In about half a century, Nevanlinna theory has been well developed and canbe used in the rsearch of the uniqueness of meromorphic functions and the complexdifferential equations and other fields. R.Nevanlinna used the first and the secondNevanlinna basic fundamental theorem he established to obtain two famousNevanlinna’s four-value theorem and Nevanlinna’s five-value theorem, which is thebasis for the uniqueness of meromorphic functions and in particular the shared valuesof meromorphic function. From the end of the1950to the begin of1960s, QiongQinglai and Yang Le, the older mathematician in our country, made some profoundresults in this respect. In1980s, famous mathematician F. Gross、G. G. Gundersen、M.Ozawa、G.Frank、 E.Mues、 N. Steinmetz、 W. Bergweiler、Hong-Xun Yi andso on obtained many important research results in the uniqueness theory ofmeromorphic function. In1995, Hong-Xun Yi solved the Gross question which wasnot solved in the past twenty years, which plays a role in the development of theuniqueness theory of meromorphic function.Recently, Yik-Man Chiang、Shao-Ji Feng、R.G.Halburd、R.J. Korhonen、I.Laineand so on, have founded Nevanlinna characteristic of difference, the first and thesecond Nevanlinna basic fundamental theorem of difference, difference analogue ofthe lemma on the logarithmic derivative and difference analogue of the Clunie lemma,To study the properties of the solutions of the difference equation and the differenceuniqueness theory to lay the foundation.In the20thcentury, the concept of meromorphic functions was originallyintroduced by French mathematician P.Montel.In rencent years, the very activeconcepts of Julia and Fatou sets in the complex analytic dynamic system are defined,which is based on the normality of a family of meromorphic functiongenerated by the iteration of meromorphic function.Many famous scholars work onthe theory of normal family of meromorphic function in the world,for example, thefamous English mathematicianW.K.Hayman,Israel mathematician L.Zalcman,Germanmathematician W.Bergweiler and so on.In our country, Zhang Qi-tai,Yang Le, WangYue-fei,Wu Jian-sheng,Gu-Yong-xing,Pang Xue-cheng,Fang Ming-liang, ChangJian-ming and others are engaged in the research of this field,the research of academicYang Le led the world.In the1990s, W.Schwick combined the normal family withuniqueness theory, in particular, in2000, Zalcman-Pang lemma was established byPang Xue-cheng and L.Zalcman which promote the research of normal families ofmeromorphic function related to public values. In China and Israel, manymathematics workers has made many important results in this respest.The present dissertation is the author’s research work under the cordial guidanceProfessor Xiao-min Li. It consists of four chapters.In chapter one, we briefly introduce some main concepts, usual notations,classical results in the value distribution theory of meromorphic functions, the resultsof difference Navenlinna theories and normal families theory.In chapter two, we study the value sharing of certain differential polynomials andtheir shifts of meromorphic functions.In chapter three, we study the result on value distribution of certain differencepolynomials of meromorphic funtions.In chapter four, we study the result on value distribution of certain differentialpolynomials and difference polynomials of meromorphic functions.