Copyright © 2011 Xiao Bing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the normality of families of holomorphic functions. We prove the following result. Let α(z),  ai(z),  i=1,2,…,p, be holomorphic functions and F a family of holomorphic functions in a domain D, P(z,w):=(w-a1(z))(w-a2(z))⋯(w-ap(z)),  p≥2. If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z),  g(z)∈F and one of the following conditions holds: (1) P(z0,z)-α(z0) has at least two distinct zeros for any z0∈D; (2) there exists z0∈D such that P(z0,z)-α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0,z)-α(z0) and that the multiplicities l and k of zeros of f(z)-β0 and α(z)-α(z0) at z0, respectively, satisfy k≠lp, for all f(z)∈F, then F is normal in D. In particular, the result is a kind of generalization of the famous Montel's criterion. At the same time we fill a gap in the proof of Theorem 1.1 in our original paper (Wu et al., 2010).