2009 Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives
Jianming Qi, Feng Lü, Ang Chen
Abstr. Appl. Anal. 2009: 1-9 (2009). DOI: 10.1155/2009/847690

## Abstract

We use the theory of normal families to prove the following. Let ${Q}_{1}(z)={a}_{1}{z}^{p}+{a}_{1,p-1}{z}^{p-1}+\cdots +{a}_{1,0}$ and ${Q}_{2}(z)={a}_{2}{z}^{p}+{a}_{2,p-1}{z}^{p-1}+\cdots +{a}_{2,0}$ be two polynomials such that $\deg {Q}_{1}=\deg {Q}_{2}=p$ (where $p$ is a nonnegative integer) and ${a}_{1},{a}_{2}({a}_{2}{\,\neq\,}0)$ are two distinct complex numbers. Let $f(z)$ be a transcendental entire function. If $f(z)$ and ${f}^{\prime }(z)$ share the polynomial ${Q}_{1}(z)\text{\,CM}$ and if $f(\text{z})={Q}_{2}(z)$ whenever ${f}^{\prime }(z)={Q}_{2}(z)$, then $f\equiv {f}^{\prime }$. This result improves a result due to Li and Yi.

## Citation

Jianming Qi. Feng Lü. Ang Chen. "Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives." Abstr. Appl. Anal. 2009 1 - 9, 2009. https://doi.org/10.1155/2009/847690

## Information

Published: 2009
First available in Project Euclid: 16 March 2010

zbMATH: 1177.30038
MathSciNet: MR2516013
Digital Object Identifier: 10.1155/2009/847690