Criteria for univalency and quasiconformal extension for harmonic mappings

Abstract

In this paper, we study the univalency and quasiconformal extension of sense-preserving harmonic mappings f in the unit disk. For f, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of f, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernández and Martín [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmüller mappings.