Abstract and Applied Analysis
Volume 2011 (2011), Article ID 373910, 8 pages
Research Article

Composite Holomorphic Functions and Normal Families

1Department of Mathematics, Xinjiang Normal University, Urumqi 830054, China
2Shaozhou Normal College, Shaoguan University, Shaoguan 512009, China
3School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received 27 March 2011; Accepted 25 July 2011

Academic Editor: Alexander I. Domoshnitsky

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.


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)),  p2. If Pwf(z) and Pwg(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 z0D; (2) there exists z0D 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 klp, 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).