Some Results On Meromorphic Functions Sharing Sets

The value distribution theory founded by Rolf Nevanlinna in the 1920's. Usually, we called Nevanlinna theory in honor of him. Nevanlinna theory can be seen the most important achievements in the preceding century to understand the properties of meromorphic functions. This theory is composed of two main theorems, which are called Nevanlinna's first and second main theorems that had been significant breakthroughs in the development of the classical function theory, since the later generalizes and extends the Picard's first theorem greatly, and hence it denoted the beginning of the theory of meromorphic functions. Moreover, Nevanlinna theory and its extensive has numerous applications in some fields of mathematics, for example, potential theory, complex differential and difference equations, normal family, several complex variables and so on.Ncvanlinna's theory is very useful in the meromorphic uniqueness theory. The uniqueness theory of meromorphic functions mainly studies conditions un-der which there exists essentially only one function satisfying these conditions. It is well known that any polynomial is determined by its zero points(the set on which the polynomial take zeros) except for a non-constant factor, but it is not true for transcendental entire function. Therefore, how to uniquely determine a meromorphic function is interesting and complex. In this field, the value distri-bution theory established by R. Nevanlinna becomes naturally the main tool for the studies. Early, Nevanlinna himself proved that any non-constant meromor-phic function can be uniquely determined by five values. In other words, if two non-constant meromorphic functions f and g take the same five values at the same points, then f≡g. Certainly, the number of five can be reduced when more conditions been added. The book [45] introduced various results and meth-ods for studying the uniqueness theorems of mcromorphic functions in different situations. One important aspect of the uniqueness theory is on uniqueness of meromorphic functions having shared sets. Recent years there is one great im-provement in this field is the work by Yi, who solved a well-known problem posed by Gross.The present thesis involves some results of the author on a result of H. Fujimoto, unicity of meromorphic functions due to a result of Yang-Hua,the unicity of meromorphic functions in Cm and an IM-shared unique range set for mcromorphic functions with 10 elements. It consists of five parts and the matters are explained as below.In Chapter 1, we introduce the general background of Ncvanlinna Theory and the development of uniqueness theory of meromorphic functions.In Chapter 2, we study such a problem:let P(ω) be a uniqueness polynomial of degree q without multiple zeros whose derivative has mutually distinct k zeros di with multiplicities ql for l=1,2,..., k respectively, and let S:={a1, a2,aq} be the zero set of P(ω). Under the assumption that P(dls)≠P(dlt) (1≤ls6/7, there exists a set S(?)C consisting of 10 elements such that the condition E(S, f)=E(S,g) implies f(?)g.