Uniqueness Of Meromorphic Functions Concerning Sharing One Small Function

In the 1920's, R. Nevanlinna,a famous Finnish mathematician,introduced the characterristic functions of meromorphic functions,and established two fundamental theorems of meromorphic functions,then the study of the value distribution theory of modern times started.The uniqueness theory of meromorphic functions is an impor-tant subject in value distribution theory. in In the past five decades or more,many mathematics scholars pay great attentions to this field which would always be based on Nevanlinna's theory.For example,E. Mues, F. Gross, G. G. Gundersen, G. Frank, N. Steinmetz, W. Bergweiler, I. Laine, C. C. Yang, Q. L. Xiong, L. Yang, H. X. Yi and Y. X. Gu had done a lot of work on the research of the uniqueness theory.The uniqueness of meromorphic functions concerning sharing one value or one small func-tion is an important part of the uniqueness theory and an subject of great value.In this paper, we mainly study the uniqueness of meromorphic functions concerning sharing one small function and discuss the uniqueness problems of a meromorphic function and its homogeneous differential polynomials with multiplicity<≤k sharing one small function.This paper is divided into five chapters:In chapter one, we mainly introduce the development of Nevanlinna theory and some main concepts,usual notations and classical results of it.In chapter two, as an special case of the uniqueness problems on a meromorphic function sharing one small function with its differential polynomials, we are mainly concerned with the problem of Hayman from the view of sharing one small func-tion.We improve some former results given by Mues-Steinmetz,W.C.Lin, M.L.Fang and J.L.Zhang.In chapter three, we are mainly devoted to the uniqueness problems on a mero-morphic function sharing one small function with its homogeneous differential poly-nomials,which generalize the results of Kit-wing Yu, I.Lahiri and L.Z.Yang. In chapter four,going on the study of chapter three, we mainly discuss the uniqueness problems of a meromorphic function and its differential polynomials with multiplicity≤k sharing one small function.The conclusion extend and improved the result ahead.In chapter five,taking out the limitation of homogeneous, we consider over the uniqueness problems on a meromorphic function sharing one small function with its differential polynomials in a general way,which generalize the results of Li.Pei.Liu.