Arcwise Connected Domains, Quasiconformal Mappings, and Quasidisks

Abstract

We prove that a homeomorphism $f:R^2\to R^2$ is a quasiconformal mapping if and only if $f(D)$ is an arcwise connected domain for any arcwise connected domain $D\subseteq R^2$, and $D$ is a quasidisk if and only if both $D$ and its exterior $D^{\mathrm{*}}=R^2\setminus \overline{D}$ are arcwise connected domains.