2020/09/30 by Iason Efraimidis, Efraimidis, Iason
Mathematics · #30C55 #30C62 #31A05 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2009.14766
openalex publication_date 2020/09/30 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
If \Ω is a simply connected domain in \ mathbb C then,\naccording to the Ahlfors-Gehring theorem, \Ω is a quasidisk if and only\nif there exists a sufficient condition for the univalence of holomorphic\nfunctions in \Ω in relation to the growth of their Schwarzian derivative.\nWe extend this theorem to harmonic mappings by proving a univalence criterion\non quasidisks. We also show that the mappings satisfying this criterion admit a\nhomeomorphic extension to \ mathbb C and, under the additional\nassumption of quasiconformality in \Ω, they admit a quasiconformal\nextension to \ mathbb C.\n The Ahlfors-Gehring theorem has been extended to finitely connected domains\n\Ω by Osgood, Beardon and Gehring, who showed that a Schwarzian criterion\nfor univalence holds in \Ω if and only if the components of\n\∂\Ω are either points or quasicircles. We generalize this theorem\nto harmonic mappings.\n