Normality Of Two Families Concerning Shared Values And Hyperbolic Metric

This paper consists of three parts which discuss three topics respectively. Prom the viewpoint of two families concerning shared values (or hyperplanes) and hyper-bolic metric, we study the normality of family of meromorphic functions.In the first part, we consider the normality of two families on sharing values, which generalizes the normality of one family on sharing values. Our work is based on the research of Xuecheng Pang and Xiaojun Liu etc on two families concerning shared values. In chapter 3, we discuss the normality of two families concerning shared function and obtain some related results. The major results are: Thoerem 3.2 Let a(z)≠ 0, ∞ be a meromorphic function on a domain DcC. Set F be a family of holomorphic functions on D, and for every f∈F satisfying the following conditions:Suppose F1 and F2 be two subsequences of F and F2be normal on D. If for each f∈F1, there exists g∈F2 such that f(z) and g(z) share the value 0, then F1 is Thoerern 3.3 Let a(z)≠ 0,∞ be a meromorphic function on a domain D∈C. Set F be a family of meromorphic functions on D, and for every f∈F satisfying the following conditions: Suppose F1 and F2 be two subsequences of F and F2 be normal on D. If for each f∈F1, there exists g∈F2 such that f(z) and g(z) share the value 0, then F1 is normal on D.In chapter 4, we study the normality of family of holomorphic curves concerning shared hyperplanes. We derive the following two theorems:Theorem 4.1 Let F∈H(D;PN(C)), q (≥ 2n+1) be a positive integer. Set H1,…, Hq be q hyperplanes in P"(C) located in general position. And suppose that for each f, g∈F, f and g share Hj (j= 1,…, q) on D, then F is normal on D.Theorem 4.2 Let F, Gf∈H(D; Pn(C)), q (≥ 3n+1) be a positive integer. And suppose the following three conditions are satisfied:(i). For each f∈F, there exist g∈G and q hyperplanes H1,…, Hq,f(which may depend on f) such that f and g share Hj,f(j= 1,...,q) on D.Then F is a normal family on D.In the second part, we investigate the normality of family of meromorphic functions from the viewpoint of metric. The Euclidean metric and spherical metric are replaced by the hyperbolic metric, and we get some sufficient and necessary conditions which can determine a given family of meromorphic functions is normal or not. Two mainly theorems we get in this part are:Thoerem 5.1 Let F be a family of holomorphic functions on the unit disc Δ. ThenThoerem 5.2 Let F be a family of meromorphic functions on the unit disc Δ. Then F is normal on Δ if and only if the following alternative holds:We give an application about the theorem 5.2. This is the first time that one discusses the normal family under the hyperbolic metric.