Uniqueness Problems And Singular Directions Of Meromorphic Functions

In1925, R.Nevanlinna established two fundamental theorems of meromorphic functions and the study of the value distribution theory of modern times started. Several decades, the further and new development of the value distribution of meromorphic functions would always be based on Nevanlinna’s theory. In this paper, based on Nevanlinna’s theory, some problems in the uniqueness theory and singular directions of meromorphic functions are studied. Also, the whole paper is divided into three chapters.The first chapter of the paper starts from the preliminary and some basic theorems and results in Nevanlinna’s theory of meromorphic functions.In Chapter2, some uniqueness problems of entire functions concerning their differential polynomials sharing the value1with finite weight were investigated and some theorems were obtained as follows:Let f and g be two nonconstant entire functions, and let n,k be two positive integers. If(fn)(k) and (gn)(k) share (1,l), and if one of conditions (i)l=0and n>5k+7;(ii)l=1and n>4k+9/2;(iii) l=2and n>3k+4; satisfied, then either f=c1ecz, g=c2e-cz where c,c1and c2are three costants satisfying (-1)k (c1c2)n (nc)2k=1or f(z)=tg(z) for a constant t such that t"=1. These results in this paper complement and improve some results given by some scholars.In Chapter3, the definitions of some singular directions of meromorphic functions and the relations and the development of these singular directions are introduced.