Quasi-isometricity and Equivalent Moduli of Continuity of Planar 1/|\omega|^2-Harmonic Mappings

Abstract

In this paper, we prove that 1/|\omega|^2-harmonic quasiconformal mapping is bi-Lipschitz continuous with respect to quasihyperbolic metric on every proper domain of C\{0}. Hence, it is hyperbolic quasiisometry
in every simply connected domain on C\{0}, which generalized the result obtained in [14]. Meanwhile, the equivalent moduli of continuity for 1/|\omega|^2-harmonic quasiregular mapping are discussed as a by-product.