Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains

Abstract

If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension to $\overline{\mathbb C}$. The Ahlfors-Gehring theorem has been extended to finitely connected domains $Ω$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $Ω$ if and only if the components of $\partialΩ$ are either points or quasicircles. We generalize this theorem to harmonic mappings.