Copyright © 2010 Qifeng Wu 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

A result of Hinchliffe (2003) is extended to transcendental entire function, and an alternative proof is given in this paper. Our main result is as follows: let α(z) be an analytic function, ℱ a family of analytic functions in a domain D, and H(z) a transcendental entire function. If H∘f(z) and H∘g(z) share α(z) IM for each pair f(z),g(z)∈ℱ, and one of the following conditions holds: (1) H(z)−α(z0) has at least two distinct zeros for any z0∈D; (2) α(z) is nonconstant, and there exists z0∈D such that H(z)−α(z0):=(z−β0)pQ(z) has only one distinct zero β0, and suppose that the multiplicities l and k of zeros of f(z)−β0 and α(z)−α(z0) at z0, respectively, satisfy k≠lp, for each f(z)∈ℱ, where Q(β0)≠0; (3) there exists a z0∈D such that H(z)−α(z0) has no zero, and α(z) is nonconstant, then ℱ is normal in D.