Several Problems In Relation To Normal Famly Theory Of Meromorphic Functions

In 1920s,R.Nevanlinna introduced the characteristics functions of meromorphic functions and gave the famous Nevanlinna theory. This theory is the most important achievements in twenty centries . For over a half century , Nevanlinna theory has been well developed and can be used in the research of meromorphic functions. Especially , there have been losts of celebrated results in the normal famlity theory, the complex differential equations, the uniqueness of meromorphic functions and other research fields, but there many open questions have been no answers, for instance, the Gross problem, W. Bergweiler conjecture, the Yi-Yang prolem etc, which are inrerested by many complex experta. In this thesis, by using Nevanlinna theory, we gave some researches on the theory of meromorphic function under the guidance of Professor Li jiangtao.In Section 1 , we introduce some fundamental results, some notations.In Section 2 , we study the normality of a famility of holomorphic functions .a general normal criterion is obtained, which improves the results given by Li JT and Yi HX [ 26].That is , f~ (k) is replaced by a differential polynomial (and M the result is obtained. We obtain the follow result.Theorem 1 Let F be a families of holomorphic functions in a domain D. m,q, k be positive intrgers,and let H( f , f`,…, ( f~ (k))) be a differential polynomial which satisfies r~+_H >0 , a(z), b(z)≠0, c(z)≠0 be some analytic functions in D. If for each f∈F, the zeros of f have multiplicity at least k, and Then F is noemal in D.In Section 3, In this paper,we study normality of a familiy of meromophic functions and general criteria for normality of a familiy of meromophic functions with multiple zeros concerning shared values are obtained..Some example are provided to show the result is sharp. We obtain the therom3.1.2-3.1.6.