Set shared by an entire function with its $k$-th derivatives using normal families

Abstract

In this paper, we study a problem of non-constant entire function $f$ that shares a set $\mathcal{S}=\{a,b,c\}$ with its $k$-th derivative $f^{(k)}$, where $a,b$ and $c$ are any three distinct complex numbers. We have found a gap in the statement of the main result of \textit{Chang-Fang-Zalcman} \cite{Cha & Fan & al-ADM-2007} and with some help of the method used by \textit{Chang-Fang-Zalcman}, we have generalized the result of \textit{Chang-Fang-Zalcman} in a more compact form. As an application, we generalize the famous Br$\ddot{u}$ck conjecture \cite{Bru-1996} with the idea of set sharing.