The Uniqueness Of Meormorphic Functions Sharing Three Values And Sharing One Value With Their Derivatives

In 1920s, R.Nevanlinna, a famous Finnish mathematician, introduced the characteristic functions of meromorphic functions and so founded the value distribution theory of meromorphic function. The theory is surely one of the most important achievements in mathematics in the 20th century because not only it is the basis of modern meromorphic function theory, but also it has quite effect on the development of many other mathematical branches, and on the intersection among them. In about half a century, Nevanlinna theory has been well developed and can be used in the rsearch of the complex differential equations, the uniqueness of meromorphic fun-ctions and other fields. Eespecially, the Nevanlinna theory supplies a very powerful tool to the research of the global analytic solution of comp-lex differential equations(see e.g.[4])and also makes the research field more active and popular after it was successfully applied to get insight into the solutions of complex differentail equations. On the other hand, in the uniqueness of meromorphic functions field, Nevanlinna studied the conditions with which a meromorphic function can be determined by the theory just established in 1929(see e.g.[6]) by himself, and obtained two celebrated uniqueness theorems for meromorphic functions, which are usually called Nevanlinna's five-value theorem and Nevanlinna's four-va-lue theorem. This launched the investigation of uniqueeness theory of meromorphic functions and in part icular the shared values of meromorphic functions.In the past half a century, in China,Japan,Germany,England,the former Soviet Union and the United States, many mathematicians devoted themselv-es to the research into the uniqueness theory of meromorphic functions,and thus it became an important branch in the complex analysis field, even up till now, which is still active. Its distinctive mode of thinking and research skills have provided important enlightenment and reference for the survey of uniqueness problems and related topics in other mathematical branches, such as algebroidal function, meromorphic function in Non-A rchimedean field and even meromorphic mapping on general manifolds.In the past two decades, Professor Hong-Xun Yi did many original work(see e.g. [2] [7]) on the uniqueness theory of meromorphic functions, and thus drew interest of mathematicians in different countries, which well impelled the development of the uniqueness theory of meromorphic functions and made it more perfect gradually. He has contributed greatlly to the promotion of the international position of China in this research area.Professor Xiao-Min Li is more active in the research of uniqueness theory of meromorphic functions, whose researches are readed by mathemat-icians from different countries. Furthermore, he also has obtained more achievements in research of complex differential equations and normal families. For example, he has studied the Bruck conjecture and Gundersen question(see e.g.[8][9]).The present dissertation is the author' research work under the cordi-al guidance Professor***. It consists of three chapters.In chapter one, we briefly introduce some main concepts, usual notati-ons and classical results with this dissertation in the value distribution theory of meromorphic functions.The research on uniqueness of meromorphic functions dealing with shared values has been studied firstly by R.Nevanlinna who obtained unici-ty theorem for three meromorphic functions sharing three values CM. Then, mathematicians from different countries, such as Hong-Xun Yi, Xiao-Min Li, Ueda, Brosch, K.Tohge, Jank-Volkmann and E.Mues and so on, obtained many unicity theorem for periodic or even functions, multiplicities and uniqueness, deficiency and uniqueness, hyper-order and uniqueness and the uniqueness related to the solution of the differential equation. In chapt-er three, we deal with the problem of uniqueness and weighted sharing of two meromorphic functions under the condition of their derivatives sharing a small rational function CM and obtain the following result which is a provement of K.Tohge, Hong-Xun Yi and Xiao-Min Li.The first two who made studies in the shared value of entire functions and their derivatives seems L.A.Rubel and C.C.Yang. Later on, some foreign complex analysis specialists, such as E.Mues, G.Frank, N.Steinmetz, G.G.Gundersen, G.Jank and L.Volkmann as well as few Chinese scholars, continuously go deep into the subject in different directions respective-ly, and have obtained a large number of excellent results. However, there remain some open questions. Moreover, in 1992, W.Schwick(see e.g.[24]) discovered that there existed a close relation between the normality of a family of entire functions and the property whether every function in the family shares value with its derivative or not. We believe that the above discovery will be surely to advance the research into shared value problem for entire functions.In chapter two, we deal with the uniqueness problem of meromorphic functions that share three values CM.The results in this chapter improve and supplement those given by H.X.Yi,X.M.Li and H.X.Yi and other authors.Now we turn to state the main results of the thesis.Theorem 1 Let f and g be two distinct nonconstant meromorphic functions sharing 0,1,∞CM,and let P(z) be a nonconstant polynomial such that (P'(z))'≠1, where t is an arbitrary positive integer.If then f=g.Theorem 2 Let f and g be two distinct nonconstant meromorphic functions sharing 0,1,∞CM,and let P(z) be a nonconstant polynomial.Ifσ(f)<∞and Then f=gIn chapter three, we study uniqueness problem for entire functions that share two small functions with derivatives, and obtain a main uniqueness theorem which is a provement of results given by L.A.Rubel, G.Frank, Jia-Hua Zheng, Shu-pei Wang and Lian-Zhong Yang. Theorem 3 Let f be a nonconstant meormorphic function,suppose that k is a positive interger,and n is a positive interger with 2n>k+3+(?).If fn and fn-1 f(k) share 1 CM,then f=f(k)Theorem 4 Let f be a nonconstant meormorphic function,suppose that k is a positive interger, and n is a positive interger with 2n>3k+8+(?).If fn and fn-1 f(k) share 1 IM,then f=f(k).