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. 1. Introduction Let and be two domains of hyperbolic type in the complex plane . A sense-preserving homeomorphism of onto is said to be a -harmonic mapping if it satisfies the Euler-Lagrange equation where and is a smooth metric in . If is a constant then is said to be euclidean harmonic. A euclidean harmonic mapping defined on a simply connected domain is of the form , where and are two analytic functions in . For a survey of harmonic mappings, see [1–3]. In this paper we study the class of -harmonic mappings. This class of mappings seems very particular but it includes the class of so-called logharmonic mappings. In fact, a logharmonic mapping is a solution of the nonlinear elliptic partial differential equation where is analytic and (see [4–6] for more details). By differentiating (1.2) in , we have that Hence, it follows that a logharmonic mapping is a -harmonic mapping. If a -harmonic mapping also satisfies the condition that holds for every , then it is called a -harmonic -quasiconformal mapping (for simplicity, a harmonic quasiconformal mapping or H.Q.C mapping), where . Let denote the hyperbolic metric of a simply connected region with gaussian curvature ？4. For a harmonic quasiconformal mapping of onto , we call the quantity the hyperbolically partial derivative of . If is a harmonic quasiconformal mapping of onto and is a conformal mapping of onto then is also a harmonic quasiconformal mapping. We have where . Hence, we always fix the domain of a harmonic quasiconformal mapping to be the unit disk when studying its hyperbolically partial derivative. The hyperbolic distance between and is defined by , where runs through all rectifiable curves in which connect and . A harmonic quasiconformal mapping of onto is said to be hyperbolically -Lipschitz if The constant is said to be the hyperbolically Lipschitz coefficient of . If there also exists a constant such that then is said to be hyperbolically -bi-lipschitz. We also call the array the hyperbolically bi-lipschitz coefficient of . Under differently restrictive conditions of the ranges of euclidean harmonic quasiconformal mappings, recent papers [7–13] obtained their euclidean Lipschitz and
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