Conformal Mappings of Domains Satisfying a Wedge Condition

Abstract

A plane Jordan curve $\Gamma$ satisfies an interior (exterior) wedge condition if for some $\alpha \in (0,1)$ there is a fixed wedge of opening $\alpha \pi$ such that for any $\omega \in \Gamma$ one may place a wedge inside (outside) $\Gamma$ with vertex at $\omega$. Let $f$ be a conformal mapping of the disk $D$ onto the interior of $\Gamma$. We establish Hölder continuity of $f({f^{ - 1}})$ on $\partial D(\Gamma )$ with best possible exponents in terms of $\alpha$.