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

Iason Efraimidis¹

¹Texas Tech University, Department of Mathematics and Statistics

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 1123-1134

Résumé

If $\Omega$ is a simply connected domain in $\overline{\mathbf C}$ then, according to the Ahlfors-Gehring theorem, $\Omega$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $\Omega$ 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 $\overline{\mathbf C}$ and, under the additional assumption of quasiconformality in $\Omega$, they admit a quasiconformal extension to $\overline{\mathbf C}$. The Ahlfors-Gehring theorem has been extended to finitely connected domains $\Omega$ by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion for univalence holds in $\Omega$ if and only if the components of $\partial\Omega$ are either points or quasicircles. We generalize this theorem to harmonic mappings.  

Détail

Keywords: Harmonic mappings, Schwarzian derivative, univalence criterion, quasiconformal extension