ENTIRE FUNCTIONS AND THEIR HIGHER ORDER DIFFERENCES

Abstract

In this paper, we prove that for a transcendental entire function $f(z)$ of finite order such that $\lambda(f-a(z))\lt \sigma(f)$, where $a(z)$ is an entire function and satisfies $\sigma(a(z))\lt 1$, $n$ is a positive integer, if $\Delta_{\eta}^nf(z)$ and $f(z)$ share entire function $b(z)\,(\,b(z)\not\equiv a(z))$ satisfying $\sigma(b(z))\lt 1$ CM, where $\eta\,(\in\mathbb{C})$ satisfies $\Delta_{\eta}^nf(z)\not\equiv 0$, then $$f(z)=a(z)+ce^{c_1z},$$ where $c,\,c_1$ are two nonzero constants.