Abstract
We study the class of -harmonic -quasiconformal mappings with angular ranges. After building a differential equation for the hyperbolic metric of an angular range, we obtain the sharp bounds of their hyperbolically partial derivatives, determined by the quasiconformal constant . As an application we get their hyperbolically bi-Lipschitz continuity and their sharp hyperbolically bi-Lipschitz coefficients.
Highlights
Let Ω and Ω be two domains of hyperbolic type in the complex plane C
A euclidean harmonic mapping defined on a connected domain is of the form f h g, where h and g are two analytic functions in Ω
In this paper we study the class of 1/|w|2-harmonic mappings
Summary
Let Ω and Ω be two domains of hyperbolic type in the complex plane C. In this paper we study the corresponding question for the class of 1/|w|2-harmonic quasiconformal mappings To this question, Examples 5.1, and 5.2 show that if the metric ρ is not necessary to be smooth in the range of a ρ-harmonic quasiconformal mapping f, f generally does not need to have euclidean and hyperbolically Lipschitz continuity even if its range is convex. In this paper we will study the hyperbolically Lipschitz or bi-lipschitz continuity of a 1/|w|2-harmonic quasiconformal mapping with an angular range and its sharp hyperbolically Lipschitz coefficient determined by the constant of quasiconformality. The key of this paper is to build a differential equation for the hyperbolic metric of an angular domain, which is different for using a differential inequality when we studied the class of euclidean harmonic quasiconformal mappings in 14. In order to show the sharpness of Theorems 3.1 and 4.1, we present two examples satisfying that the inequalities 3.2 no longer hold for two classes of 1/|w|2-harmonic quasiconformal mappings with nonangular ranges see Examples 5.4 and 5.5
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