On entire functions that share two small functions with their derivative

Abstract

In this paper, we study the relationship between two entire functions sharing two pairs of small functions IM with their derivatives. For such a function ƒ and a positive integer k, we establish a linear relationship between ƒ and ƒ(k) which generalizes a result due to Chen (Abstr. Appl. Anal.. Art. 601507. 1-9, 2014). Examples have also been given in support of the existence of the result and the sharpness of some conditions involved in it. Furthermore, we also study a uniqueness result concerning ƒ and ƒ(k) sharing two small functions IM.