Uniqueness Of Meromorphic Functions And Differntial Polynomials That Share One Value

In 1920s, R.Nevanlinna introduced the characteristics functions of meromorphic functions and founded the famous Nevanlinna theory,which is one of the most important achievements on mathematics in twenty centuries. In the past half a century, Nevanlinna theory has been well developed and can be used in the research of meomorphic functions. Especially, there have been losts of celebrated results in the uniqueness theory. For instance, Yang Le,Yi Hongxun, Yang Chongjun etc , who did many original work on the uniqueness theory. We gave some researches on the theory of meromorphic function base on Nevanlinna theory.In Section 1, we introduce some fundamental results, and some notations, the main results of this paper and the background of theorems.In Section 2, we study the uniqueness of meromorphic functions and entire functions with a weighted sharing method, and obtain some uniqueness theorems which improve and generalize the related results of Liu Lipei. This is, let f ( z ) and g ( z ) be two nonconstant transcendental meromorphic functions, and let n , l , m and k be four positve integers. If , and one of the following condtions holds,max max and satisfy the algebraic equationwhere In the same wav,en fand gbetwo nonconstant entire functions,we ProofTheory 2.In Section3,we study the uniqueness of rneromorphic functions concerningdifferential Polynomials and sharing vafues.