Uniqueness Theorems on Difference Monomials of Entire Functions

Abstract

The aim of this paper is to discuss the uniqueness of the difference monomials $f^{n}f(z+c)$. It assumed that $f$ and $g$ are transcendental entire functions with finite order and $E_{k)}(1,f^{n}f(z+c))=E_{k)}(1,g^{n}g(z+c))$, where $c$ is a nonzero complex constant and $n$, $k$ are integers. It is proved that if one of the following holds (i) $n\ge 6$ and $k=3$, (ii) $n\ge 7$ and $k=2$, and (iii) $n\ge 10$ and $k=1$, then $fg=t_1$ or $f=t_{2}g$ for some constants $t_2$ and $t_3$ which satisfy $t_2^{n+1}=1$ and $t_3^{n+1}=1$. It is an improvement of the result of Qi, Yang and Liu.