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, $D^\vee=\{w\in\mathbf{C}:\, 1/w\in\hat{\mathbf{C}} \setminus\bar D\}$. This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.