Abstract and Applied Analysis
Volume 2009 (2009), Article ID 847690, 9 pages
doi:10.1155/2009/847690
Research Article

Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

Jianming Qi,1 Feng Lü,2 and Ang Chen1

1School of Mathematics and System Sciences, Shandong University, Jinan, Shandong 250100, China
2Department of Mathematics, China University of Petroleum, Dongying, Shandong 257061, China

Received 1 December 2008; Revised 10 April 2009; Accepted 6 May 2009

Academic Editor: Stevo Stevic

Copyright © 2009 Jianming Qi 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 use the theory of normal families to prove the following. Let Q1(z)=a1zp+a1,p−1zp−1+⋯+a1,0 and Q2(z)=a2zp+a2,p−1zp−1+⋯+a2,0 be two polynomials such that deg⁡Q1=deg⁡Q2=p (where p is a nonnegative integer) and a1,a2(a2≠0) are two distinct complex numbers. Let f(z) be a transcendental entire function. If f(z) and f′(z) share the polynomial Q1(z) CM and if f(z)=Q2(z) whenever f′(z)=Q2(z), then f≡f′. This result improves a result due to Li and Yi.