Some Normal Criteria about Shared Values with Their Multiplicity Zeros

Let be a family of meromorphic functions in the domain , all of whose zeros are multiple. Let be an integer and let , be two nonzero finite complex numbers. If and share in for every pair of functions , then is normal in . 1. Introduction and Main Results We use to denote the open complex plane, to denote the extended complex plane, and to denote a domain in . With renewed interest in normal families of analytic and meromorphic functions in plane domains, mainly because of their role in complex dynamics, it has become quite interesting to talk about normal families in their own right. We will be concerned with the analytic maps (i.e., meromorphic functions) from (endowed with the Euclidean metric) to the extended complex plane endowed with the spherically metric given by A family of meromorphic functions defined in is said to be normal, in the sense of Montel, if for any sequence , there exists a subsequence such that converges spherically locally and uniformly in to a meromorphic function or . Clearly, is normal in if and only if it is normal at every point of (see [1, 2]). Let and be two nonconstant meromorphic functions in , and . We say that and share the value in , if and have the same zeros (ignoring multiplicities). When the zeros of means the poles of (see [3]). Influenced from Bloch's principle [4], every condition which reduces a meromorphic function in the plane to a constant, makes a family of meromorphic functions in a domain normal. Although the principle is false in general (see [5]), many authors proved normality criterion for families of meromorphic functions corresponding to Liouville-Picard type theorem (see [1, 2, 6]). It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schiff [1] first proved an interesting result that a family of meromorphic functions in a domian is normal if in which every function shares three distinct finite complex numbers with its first derivative. And later, Sun [7] proved that a family of meromorphic functions in a domian is normal if in which each pair of functions share three fixed distinct valus, which is an improvement of the famous Montel's Normal Criterion [8] by the ideas of shared values. More results about normality criteria concerning shared values can be found, for instance, (see [9–12]) and so on. In 1989, Schwick [13] proved that let be a family of meromorphic functions in a domain , if for every , where , are two positive integers and , then is normal in . Recently, by the ideas of shared values, Li and Gu [14] proved the following.