Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 417480, 9 pages
Research Article

Normality of Composite Analytic Functions and Sharing an Analytic Function

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

Received 17 August 2010; Accepted 15 October 2010

Academic Editor: Manuel De la Sen

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.


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, a family of analytic functions in a domain D, and H(z) a transcendental entire function. If Hf(z) and Hg(z) share α(z) IM for each pair f(z),g(z), and one of the following conditions holds: (1) H(z)α(z0) has at least two distinct zeros for any z0D; (2) α(z) is nonconstant, and there exists z0D such that H(z)α(z0):=(zβ0)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 klp, for each f(z), where Q(β0)0; (3) there exists a z0D such that H(z)α(z0) has no zero, and α(z) is nonconstant, then is normal in D.