Continuous boundary values of conformal maps

Abstract

Let $G$ be a bounded simply connected domain in the complex plane. A point $a\in \partial G$ is said to be accessible from inside of $G$ if there is a Jordan arc $J$ such that $J\subset \bar{G}$ and $J\cap \partial G=\{a\}$. In this paper the author shows that a univalent analytic function $\psi$ from the unit disk $D$ onto $G$ extends continuously to $\bar{D}$ if and only if every $a\in \partial G$ is accessible. The main result covers a famous theorem proved by C. Caratheödory, which says that if $G$ is a Jordan domain, then $\psi$ extends to be a homeomorphism from $\bar{D}$ onto to $\bar{G}$.

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