Uniqueness And Normal Family Of Meromorphic Functions Concerning Sharing Values

The uniqueness and normal family theory of meromorphic functions concerning sharingvalues and small functions are established on the base of R. Nevanlinna value distribution the-ory, which are important components of complex analysis and especially of the meromorphicfunction theory, and are mathematical topics with rich research value.In1929, R. Nevanlinna used value distribution theory to study the uniqueness theory ofmeromorphic functions, that is under what circumstances there exists only one function sat-isfying the given conditions. We know that any polynomial is determined by its zeros and anon-constant factor, but it is not true for transcendental holomorphic and meromorphic func-tions. How to determine a meromorphic function is more complex. In more than half a century,foreign mathematicians and many domestic mathematicians have got remarkable achievementsin uniqueness theory, and made it flourish.At the beginning of the last century, P. Montel introducted the concept of normal families,he defined a set of functions with some kind of compactness as a normal family. Seeking nor-mal conditions of the family of meromorphic functions, is a subject of studies of meromorphicfunctions theory. Studies on normal families mostly follow the revelation of the Bloch law. A.Bloch had noted that, if a meromorphic function in an opening plane degenerates into a con-stant when it satisfies certain conditions, then a set of such meromorphic functions satisfyingthe conditions in a domain is a normal family. In recent years, many mathematical researchersconsidered normal families and shared values together, W. Schwick first studied in this respect,Sun Daochun, Yang Xuecheng and L. Zalcman achieved important results in this field.In this paper, we mainly introduce some research about the uniqueness and normal familiesof meromorphic functions concerning sharing values, and get some results. The full text isdivided into three chapters as follows:In Chapter Ⅰ, we outline the research background, introduce several basic results and thecommon used notation in the R.Nevanlinna value distribution theory, and give the basic conceptsand classical results in the uniqueness and normal theory of meromorphic functions.In Chapter Ⅱ, we first research the uniqueness of meromorphic functions concerning dif-ferential polynomials fnf(k)and gng(k)sharing a small function a(z)CM or a(z)IM, and obtainthe results that f≡tg, where tn+1=1; or fnf(k)gng(k)≡a(z)2, greatly improve some con-clusions of Yang[13] and Fang[14]. Then we research the case (fnf)(k)and (gng)(k)share1CM, and prove Theorem2.3and Corollary2.1.In Chapter Ⅲ, we mainly study the normality of meromorphic family F. If for any func-tions f, g∈F and any positive integers n and k, where n≥2k+4, their differential polyno- mials fn+af(k)and gn+ag(k)share b, and the zeros weight-class of f and g are no less thank+1, then F is normal.