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 with the famous
Br\"{u}ck conjecture. In addition, two propositions of infinite
products are obtained. The first one is the growth property of a
certain type infinite product. The second one is the property of
entire solutions of the differential equation which concerns
infinite products.