Entrire functions and their derivatives share two finite sets (Q1608633)

This paper is devoted to studying uniqueness of entire functions in terms of sharing finite sets. More precisely, let \(f, g\) be two non-constant entire functions, let \(n\geq5\), and \(k\geq1\) be integer and define \(S_1:=\{z| z^n=1\}\), \(S_2=\{a,b,c\}\), where \(a,b,c\in \mathbb{C}\backslash\{0\}\) are distinct and \(a^2\neq bc\), \(b^2\neq ac\), \(c^2\neq ab\). Using the notation \(E(S,f):=\bigcup_{a\in S}\{z| f(z)=a\}\), the main result reads as follows: If \(E(S_1,f)=E(S_1,g)\) and \(E(S_2,f^{(k)})=E(S_2,g^{(k)})\), then \(f\equiv g\). The proof of the main theorem and of the related consequences consists of a careful analysis of several subcases which appear by making use of the value distribution of \(f^{(k)}\), resp. \(g^{(k)}\). This analysis makes strong use of a result due to \textit{H. X. Yi} [Sci. China 24, 457--466 (1994)].