Toshiyuki SUGAWA

A self-duality of strong starlikeness


In this note, we will show that a simply connected bounded domain $D$ is strongly starlike of order $\alpha<1$ if and only if so is $D^\vee,$ where $D^v$ is the analytic inversion of the exterior of $D,$ namely, $D^v=\{w\in\mathbb{C}:\, 1/w\in\widehat{\mathbb{C}} \setminus\bar D\}.$ This fact neatly explains the relationship between some known properties of strongly starlike domains and provides several new characterizations for those domains.

submission: January 28, 2000
revision: February 3, 2000
revision: June 1, 2001
revision: November 22, 2001

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