Normal Family of Meromorphic Functions concerning Shared Values

We obtain a normal criterion of meromorphic functions concerning, shared values. Let be a family of meromorphic functions in a domain and let be positive integers. Let be two finite complex constants. If, for each , all zeros of have multiplicity at least and and share in for every pair of functions , then is normal in . This result generalizes the related theorem according to Xu et al. and Qi et al., respectively. There is a gap in the proofs of Lemma 3 by Wang (2012) and Theorem 1 by Zhang (2008), respectively. They did not consider the case of being zerofree. We will fill the gap in this paper. 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 . A family of meromorphic functions defined in is said to be normal, if for any sequence contains a subsequence which converges spherically, and locally, uniformly in to a meromorphic function or . Clearly, is said to be normal in if and only if it is normal at every point of (see [1, 2]). Let be a meromorphic function in a domain . We say that is a normal function if there exists a positive number such that for all , where denotes the spherical derivative of . Let be a domain in and let and be two nonconstant meromorphic functions in . Let and be two complex numbers. If whenever , we write that If and , we write that If , we say that and share (ignoring multiplicities) on . When means (see [2]). Influenced from Bloch's principle [3], 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 [4]), many authors proved normality criterion for families of meromorphic functions corresponding to Liouville-Picard type theorem (see [2, 5, 6]). It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [7] first proved an interesting result that a family of meromorphic functions in a domina is normal if in which every function shares three distinct finite complex numbers with its first derivative. And later, Sun [8] proved that a family of meromorphic functions in a domina is normal if in which each pair of functions share three fixed distinct values, which is an improvement of the famous Montel's Normal Criterion [9] by the ideas of shared values. More results about normality criteria concerning shared values can be found, for instance, see [10–13] and so on. In 1959, Hayman [14] proved that let be a meromorphic function in , if , where is a