Uniqueness And Normal Family Of Meromorphic Functions

In 1925, R. Nevanlinna established theorems of meromorphic functions which were called value distribution theory. This theory is still an importance branch of complex function theory. In this region, many Chinese mathematicians such as Q. L. Xiong, G. P. Li, X. T. Zhuang, L. Yang, G. H. Zhang had done a lot of works and obtained considerable original results. In recent years, many mathematicians investigate the uniqueness theory and normal family of meromorphic functions in view of shared-values. They developed new flied of the value distribution theory of meromorphic functions. In this dissertation, we study the uniquness problems on meromorphic function and normal family problems based on the previous work. This dissertation is divided into five chapters:In chapter 1, we mainly introduce the development of Nevanlinna theory and some notation, some results on the uniqueness theory of meromorphic functions. Also, we introduce the conception of normal family on meomorphic functions and some important consequences.In chapter 2, we mainly deal with the uniqueness problem on differential polynomials of meromorphic functions sharing small functions, which generalize the results of W. Lin and H. X. Yi, W. L. Xiong, Seiki Mori and so on.In chapter 3, We are mainly concerned with the uniqueness problems on differential polynomials with multiplicity sharing value, which extend some results obtained by W. L. Xiong, W. Lin and so on.In chapter 4 and 5, we discuss the normal family of meromorphic functions. Firstly, we obtain the normal criteria of meromorphic functions on some type of differential monomial sharing value. Secondly, we obtain the normal criterion of meromorphic function and its derivatives sharing set, which improved the theorem of W. Schwick. Lastly, we resolve the question posed by literature[23] (Y. X. Gu, X. C. Pang and M. L. Fang, 2007).