Some Results Of Zero-order Meromorphic Functions

In1920s, R. Nevanlinna introduced the characteristic functions of meromor-phic functions and established two fundamental theorems of meromorphic fucntions which was called Nevanlinna theory. And the theory opened up the cornerstone of the study of the value distribution theory on meromorphic functions. In the re-cent centry, many mathematicians such as W. K. Hayman, E. Muse, F. Gross, G. G. Gundersen, G. Frank, N. Steinmetz, W. Bergwiller, I. Laine, C. C. Yang, Seik-i. Mori, Q. L. Xiong, L. Yang, H. X. Yi, Y. X. Gu, H. H. Chen had done a lot of work on the research of the uniqueness theory, and obtained many excellent research results. It’s well known that numerous problems of meromorphic functions theory have been solved with the rapid development of cross subjects. Whereas, there are lots of new open questions to be posed, for instance, cross problems between differ-ence and value distribution theories, which are interested by many experts recently. In this paper, we mainly study the Navanlinna properties of composite functions of order zero, and investigate the value distribution of meromorphic functions of order zero or finite positive logarithmic order composed with polynomials. Furthermore, we consider the uniqueness of composite functions with sharing values, especially the uniqueness on q-difference.This paper is divided into four sections:In section one, we firstly introduce the basic Nevanlinna theory and some clas-sical theory of the uniqueness results, and present the extended definitions of order and measure in detail. Secondly, we introduce the Borel inequality and its extended inequality.In section two, we are mainly devoted to study the Navanlinna properties and the value distribution problems on meromorphic functions of order zero or finite positive logarithmic order which compose with polynomials. We obtain two ana-logues of the lemma on the logarithmic derivative and the second main theorem of Nevanlinna theory for a class of composite meromorphic functions. Moreover we dis-cuss the value distribution of meromorphic functions of order zero or finite positive logarithmic order which compose with polynomials.In section three, we study the uniqueness problems of meromorphic functions of order zero or finite positive logarithmic order which compose with polynomials.In section four, we pose some problems.