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

Iason Efraimidis
2021 Annales Fennici Mathematici  
If Ω is a simply connected domain in 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 C and, under the additional assumption of
more » ... ality in Ω, they admit a quasiconformal extension to 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 ∂Ω are either points or quasicircles. We generalize this theorem to harmonic mappings. λ Ω π(z) |π ′ (z)| = λ D (z) = 1 1 − |z| 2 , z ∈ D, and is independent of the choice for the covering π. The size of the Schwarzian derivative of a locally univalent holomorphic function f : Ω → C is measured by its norm, given by (2) Sf Ω = sup z∈Ω λ Ω (z) −2 |Sf (z)|.
doi:10.5186/aasfm.2021.4669 fatcat:hepfvsglyrg27opcso55gdxz4y