Several Problems In Relation To Normal Families Theorem And Uniqueness Of Meromorphic Functions

In 1920s,R.Nevanlinna introduced the characteristics functions of meromorphic functions and gave the famous Nevanlinna theory. We gave some researches on the theory of meromorphic function base on Nevanlinna theory.In Section 1-2, we introduce some fundamental results, some notations.In Section3-4, we study the uniqueness of a family of meromorphic functions .A general uniqueness theorem is obtained, which improves the results given by Fang Mingliang. That is(Th3.1) f ( z )and g ( z )be two non-constant entire functions, k, nbe two positive integers. If [ fⁿ (f² - 1)](k)and [ gⁿ (g² - 1)]^kshare 1CM, and n≥2 k+11, then f ( z)≡g(z). In this paper, we also get the result (Th4.8): Let f (z)and g (z)be two non-constant entire functions, n≥13a positive integer. If fⁿ f′and gⁿ g′share z IM, then either f ( z)≡tg(z)for a constant t such that tⁿ⁺¹ =1; or g ( z)= c2e-cz, where c₁ , c₂and c are three constants satisfying 4( c₁ c₂)^{n +1} c²=-1.In Section5, we study normality of a family of meromophic functions in relation to exceptional functions. We get two criterions for normality of a family of meromophic functions (Th5.9 and Th5.10).