Quasihyperbolic Quasi-isometry and Schwarz lemma of Planar Flat Harmonic Mappings

Abstract

A sufficient condition of a flat harmonic quasiconformal mapping to be a quasihyperbolic quasi-isometry on any subdomain of $\mathbb{C}$ is given in this paper, which generalizes the corresponding results of Euclidean and $1/|\omega|^{2}$ harmonic mappings. As an application, Schwarz lemma of flat harmonic mapping is also investigated. Besides, properties and constructions of flat harmonic mapping are obtained at the same time.