A self-duality of strong starlikeness

Abstract

In this note, we will show that a simply connected bounded domain D is strongly starlike of order α<1 with respect to the origin if and only if so is D^∨, where D^∨ is the analytic inversion of the exterior of D, namely, D^∨={w∈C : 1/w∈\hat{C}{\backslash}\bar{D}}. This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.