Several Types Of Transcendental Meromorphic Function Value Distribution Problems

Nevanlinna value distribution theory, which was one of the most significant achievements in mathematics in twentieth century, laid the foundation for modern value distribution theory of meromorphic functions. And it promoted the development of complex analysis. Under the continuous development of the Nevanlinna value distribution theory, as the main research tools of transcendental meromorphic function value distribution, nevanlinna value distribution theory has gradually become a research contents that mathematicians interested in.This paper study the value distribution of transcendental meromorphic function by using the Nevanlinna value distribution theory of meromorphic function. Section one provides an overview of the research background of this research area and research status at home and abroad, and introduces the basic knowledgement of the Nevanlinna value distribution theory as well as the common notation and the fundamental theorems. Section two and Section three study the value distribution problem of transcendental meromorphic function fm (f(k))n-φ, and extend m=1to the case of m≥2then research on it. Furthermore, the distribution properties of fm(f(k))n-φ are given. Section four investigates the value distribution problem of transcendental meromorphic function f(k)+a(f(k+1))n, then extends k=0to the case of k≥1for discussion and promotes Hayman problem widely. Section five researches the value distribution problem of transcendental meromorphic function f(z)Pn[f] and extends m=1to n≥2for discussion, and popularizes some achievements of of Zhongfa Zhang.