Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

Abstract

We use the theory of normal families to prove the following. Let $Q_1\left(z\right)=a_1{z}^p+a_{1,p-1}{z}^{p-1}+\cdots +a_{1,0}$ and $Q_2\left(z\right)=a_2{z}^p+a_{2,p-1}{z}^{p-1}+\cdots +a_{2,0}$ be two polynomials such that $\mathrm{deg}{Q}_1=\mathrm{deg}{Q}_2=p$ (where $p$ is a nonnegative integer) and $a_1,a_2\left(a_2\ne 0\right)$ are two distinct complex numbers. Let $f\left(z\right)$ be a transcendental entire function. If $f\left(z\right)$ and $f^{\prime }\left(z\right)$ share the polynomial $Q_1\left(z\right)\text{\hspace{0.17em}CM}$ and if $f\left(\text{z}\right)=Q_2\left(z\right)$ whenever $f^{\prime }\left(z\right)=Q_2\left(z\right)$, then $f\equiv f^{\prime }$. This result improves a result due to Li and Yi.