| In 1920s, R.Nevanlinna, a famous Finnish mathematician, introduced the charac- teristic 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 1929, in the uniqueness of meromorphic functions field, Nevanlinna(see e.g.[25]) studied the conditions with which a meromorphi- c function can be determined by the theory just established by himself, and obtained two celebrated uniqueness theorems for meromorphic functions, which are usually called Nevanlinna's five-value theorem and Nevanlinna's four-value theorem. This launched the investigation of uniqueness theory of meromor- phic functions and in particular the shared values of meromorphic functio- ns.Professor Hong-Xun Yi did many original work (see e.g.[10][23])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 functio- ns 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 merom- orphic functions, whose researches are readed by mathematicians 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 Brück conjecture and Gundersen question(see e.g.[44][45][46]).The present dissertation is the author' research work under the cordial guidance of Professor Xiao-Min Li. It consists of three chapters.In chapter one, we briefly introduce some main concepts, usual notations and classical results with this dissertation in the value distribution theory of meromorphic functions.In chapter two, we deal with the problem of uniqueness and weighted shari- ng of two meromorphic functions under the condition of their derivatives sharing a small rational function CM and obtain the result which is a provement of results given by K.Tohge, Hong-Xun Yi and Xiao-Min Li.In chapter three, we deal with the problem of uniqueness and weighted sharing of two entire functons under the condition of"sharing a value-pair"and obtain the following result which is a provement of results given by G.Brosch, T.C.Alzahary and Hong-Xun Yi. |
