Uniqueness Theory Of Meromorphic Functions That Share Values Whit Weight

The uniqueness theorem of meromorphic functions is an important subject in value distribution theory. Many mathematicicans such as E.Mues, F.Gross, G.G.Gundersen, G.Frank, N.steinmetz, W.Bergweiler, I.Laine, C.C.Yang, Q.L.Xiong, L.Yang, H.X.Yi and Y.x.Gu had done a lot of work on the research of the uniqueness theory. In this paper, we mainly discuss the problem of weighted sharing which is between "CM" and "IM", using the notion of weight sharing, we study the uniqueness problems on a meromorphic function and its derivative and linear differential polynomials sharing value. This paper is divided into five sections:In section one, we mainly introduce the basic Nevanlinna theory, and some results about the uniqueness theory of meromorphic functions.In section two, we use the idea of weight to study the uniqueness of meromorphic function sharing one value. The conclusion improved the result of Zhou Houqing.In section three, we mainly discuss the uniqueness of meromorphic function and its derivative sharing one value with weight and get two theorems which improved the result of Wu Guirong and Chang Zhaoguang.In section four, we study the uniqueness problems of meromorphic functions concerning linear differential polynomials that share one value, and get two conclusions which improved the results of Yang Li and Chang Zhaoguang.In section five, mainly discussed the problem of the sum of deficiencies of meromorphic functions with maximal sum of quasi-deficiencies, and improved the meromorphic function of finite to infinite, which improved the results of Niu Yingxuan and He Li. We get two conclutions of meromorphic functions with maximal sum of quasi-deficiencies.