Relationship between entire functions and their derivatives sharing small function except a set

Abstract

The paper is mainly devoted to deriving the relationship between an entire function and its derivative when they share one small function except possibly a set, which is related to the famous Br?ck conjecture. In addition, two propositions of infinite products are obtained. The first one is the growth property of a certain infinite product. The second one is the property of entire solutions of the differential equation which concerns infinite products.