A Generalization of Brück’s Conjecture for a Class of Entire Functions

Abstract

In this paper, we study the uniqueness problem of entire functions sharing polynomials IM with their first derivative. As an application, we generalize Bruck’s conjecture from sharing value CM to sharing polynomial IM for a class of functions. In fact, we prove a result as follows: Let $${a({\not\equiv} 0)}$$ be a polynomial and $${n \geq 2}$$ be an integer, let f be a transcendental entire function, and let $${F = f^n}$$ . If F and F′ share a IM, then $${f(z) = Ae^{z/n},}$$ where A is a nonzero constant. It extends some previous related theorems.