Inequality,Deficiency And Uniqueness Of Meromorphic Functions

In 1925 R.Nevanlinna introduced the analytic character and the characteristic function of meromorphic function, established two fundamental theorems, which become the basis of Nevanlinna theory, and initiated the neoteric research of value distribution theory of meromorphic function. After then, many experts and academicians do a lot of efficient research of value distribution theory of meromorphic function, and get plenty of productions.The research of inequality, deficiency and uniqueness of entire function and meromorphic function are important research areas of value distribution theory of meromorphic function, it is interesting and complicated, many experts and academicians do plenty of productions about these.Nevanlinna first established an inequality which concerns with the module distribution of function itself in 1925.Milloux established an inequality in 1940, it concerns with the module distribution of meromorphic function combining with its derivative, Milloux inequality is an extend and development for the Nevanlinna second theorem.Hayman got a very interesting inequality in 1959, it concerns with the module distribution of meromorphic function combining with its derivative, it can only use two counting functions to bound the characteristic function. On the problems of reducing inequality coefficients and deficiency sum, there are important Yang Le inequality and deficiency sum theorems. Satisfying what condition, meromorphic function can be determined uniquely, Nevanlinna established the five values theorem firstly, many experts and academicians get plenty of uniqueness productions about these.By using the Nevanlinna's basic theory, this dissertation studies and discusses the problems of inequality, deficiency and uniqueness of entire function and meromorphic function. First of all this dissertation briefly presents Nevanlinna theory, some important theorems and the marks in common use, and some research productions of relational areas, also narrates the dissertation's main results in brief, then narrates the relational lemmas and proof of results. This dissertation changes the constants of the counting functions of precise Yang Le inequality to polynomial firstly, then gets the corresponding deficiency sum; besides this dissertation studies and discusses uniqueness problems of meromorphic function from two aspects, gets the uniqueness of transcendental entire function sharing one value with weight and the uniqueness of meromorphic function and its differential polynomial sharing one small function, the results obtained make complement and generalization to the present conclusions.