Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains
| Autor: | Efraimidis, Iason |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | If $\Omega$ is a simply connected domain in $\overline{{\mathbb C}}$ 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 $\overline{{\mathbb C}}$ and, under the additional assumption of quasiconformality in $\Omega$, they admit a quasiconformal extension to $\overline{{\mathbb C}}$. 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. Comment: 14 pages |
| Databáze: | arXiv |
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