Normal Families Of Meromorphic Functions Concerning Shared Values Or Shared Sets With Their Derivatives

In1920s, R.Nevanlinna, a famous Finnish mathematician, establishedNevanlinna theory, which is called the first and the second Nevanlinna basicfundamental theorem. The theory is surely one of the most important achievement inmathematics in the20thcentury and is the important tool of theoretical researchcomplex. In about half a century, Nevanlinna theory has been well developed and canbe used in the research of the complex differential equation, normal families ofmeromorphic function and other fields. The research of the uniqueness theory ofmeromorphic function involving public values originated in P. Montel some work, P.Montel introduced the concept of normal family. He firstly connected normality of afamily of the function with the problem about taking on values of the function, this isthe classical Montel normal rule. Using the Nevanlinna theory made the research onthe theory of normal family reach its peak. Involved in the situation of the family ofmeromorphic function appeared the famous Marty normal rule. In the early1990s, W.Schwick studied problems about normal families of meromorphic functions relatingnormal families and uniqueness, playing a role in promoting for the development ofthe normal family of meromorphic functions.In recent years, China’s famous mathematician Yang Le、Yong-Xing Gu、Xue-Cheng Pang et al.’s researches in the front rank in the world. At the beginning ofthis century, Xue-Cheng Pang and Israel mathematician L. zalcman got thePang-Zalcman lemma, injecting new vitality in the research for involving values ofthe theory of normal family of meromorphic functions.From the end of the1950to the beginning of1960s, Qiong Qinglai and ZhuangQitai, Yang Le and Guang-Hou Zhang, the older mathematician in our country, madesome profound results in this respect. From1980s, famous mathematician L.Zalcman、W.Schwick、W.Bergweiler、M.Essen、Xue-Cheng Pang、Ming-Liang Fang、Jian-Ming Chang、Hong-Xun Yi and so on obtained many important research resultsin the normality of meromorphic function.In1995, Hong-Xun Yi solved the Gross question which was not solved in the pasttwenty years. which plays a role in the development of the uniqueness theory ofmeromorphic function(see e.g.[4][6]). He did many original work on the uniquenesstheory of meromorphic functions, and thus drew interest of mathematicians indifferent countries, which well impelled the development of the normality of meromorphic functions and made it more perfect gradually. He has contributedgreatlly to the promotion of the international position of China in this researcharea.Professor Xiao-Min Li is more active in the research of normality ofmeromorphic functions, whose researches are readed by mathematicians fromdifferent countries. Furthermore, he also has obtained more achievements in researchof complex differential equations and uniqueness theory of meromorphic functions.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, classicalresults in the normal families of meromorphic functions and the results of differenceNavenlinna theories.In chapter two, we study the normality of a family of meromorphic functionssharing a set consisting of three distinct finite values, and the normality of a family ofholomorphic functions sharing a set consisting of two distinct finite values. Finally,we study the normality of meromorphic functions sharing a nonzero holomorphicfunction with their derivatives. Now we turn to state the main results of the thesis:In chapter three, we study the normality of families of meromorphic functionssharing two finite nonzero values with their derivatives. Then we improve and extendthe theorems in Fang-Zalcman[38]and in Chang[37].In chapter four, we study the normality of families of meromorphic functionssharing a set consisting of finite values with their derivatives. Then we improve andextend the results in Liu-Pang[46]and in Liu-Chang[47].