Abstract and Applied Analysis
Volume 2009 (2009), Article ID 847690, 9 pages
doi:10.1155/2009/847690
Uniqueness of entire functions sharing polynomials with their derivatives
Jianming Qi1
, Feng Lü2
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
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 degQ1=degQ2=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.