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

Författare

  • Iason Efraimidis Texas Tech University, Department of Mathematics and Statistics

Nyckelord:

Harmonic mappings, Schwarzian derivative, univalence criterion, quasiconformal extension

Abstract

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 quasiconformality 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.

 

Sektion
Articles

Publicerad

2021-09-13

Referera så här

Efraimidis, I. (2021). Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains. Annales Fennici Mathematici, 46(2), 1123–1134. Hämtad från https://afm.journal.fi/article/view/111233