Uniqueness Of Meromorphic Functions Concerning Weakly-weighted Sharing Small Functions

The uniqueness theory of meromorphic functions is an important subject in value distribution theory. Many mathematicians such as 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. In this paper, we mainly introduce the definition of weakly-weighted sharing which is between "CM" and "IM". Using the notion of weakly-weighted sharing, we study the uniqueness problems on a meromorphic function and its derivative or homogeneous differential polynomials sharing small functions, the uniqueness problems on some certain types of differential polynomials sharing one small function. This paper is divided into four sections:In section one, we mainly introduce the development of Nevanlinna theory and the uniqueness theory of meromorphic functions, and some results about the uniqueness theory of meromorphic functions.In section two, we mainly deal with the uniqueness problems on a meromorphic function and its kth order derivative f^(k) weakly-weighted sharing one small function. As consequences, we are able to answer four questions posed by Kit-wing Yu. Our results sharpen the results of Kit-wing Yu, I. Lahiri and A. Sarkar, L. P. Liu and Y. X. Gu.In section three, we are mainly concerned with the uniqueness problems of a meromorphic function and its homogeneous differential polynomials weakly-weighted sharing two small functions. We obtain two theorems, which improve some former results given by L. A. Rubel, C. C. Yang and X. M. Li.In section four, we are mainly devoted to the uniqueness problems on some certain types of differential polynomials weakly-weighted sharing one small function, which generalize the results of M. L. Fang, H. X. Yi, W. C. Lin, W. L. Xiong, Seiki Mori. Moreover, the example shows the results are sharp.