#### 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.

#### 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 *bi-Lipschitz* continuity. In [8], Kalaj obtained the following.

Theorem A. *Let and be two Jordan domains, let and let be a euclidean harmonic quasiconformal mapping. If and , then is euclidean Lipschitz. In particular, if is convex, then is euclidean bi-lipschitz.*

Recently, the hyperbolically Lipschitz or *bi-lipschitz* continuity of euclidean harmonic quasiconformal mappings also excited much interest (see [14–17]). In [14], Chen and Fang proved the following.

Theorem B. *Let be a euclidean harmonic -quasiconformal mapping of onto a convex domain . Then is hyperbolically -bi-lipschitz.*

Theorems A and B tell us that an euclidean harmonic quasiconformal mapping with a convex range has both euclidean and hyperbolically *bi-lipschitz* continuity. Naturally, we want to ask whether a general -harmonic quasiconformal mapping also has similar Lipschitz or *bi-lipschitz* continuity. In this paper we study the corresponding question for the class of -harmonic quasiconformal mappings.

To this question, Examples 5.1, and 5.2 show that if the metric is not necessary to be smooth in the range of a -harmonic quasiconformal mapping , then generally does not need to have euclidean and hyperbolically Lipschitz continuity even if its range is convex. Hence, we only consider the case that is smooth, that is, does not vanish in the range of a -harmonic quasiconformal mapping in this paper. Kalaj and Mateljević (see Theorem 4.4 of [18]) showed the following.

Theorem C. *Let be analytic in and a -harmonic quasiconformal mapping of the domain onto the Jordan domain . If , then is euclidean Lipschitz.*

Let be equal to , where . If the closure of the range does not include the origin, then is finite. So by Theorem C a -harmonic quasiconformal mapping with such a range has euclidean Lipschitz continuity. Example 5.3 shows that if the origin is a boundary point of then a -harmonic quasiconformal mapping does not need to have euclidean Lipschitz continuity. However, Example 5.3 also shows that there is a different result when we consider its hyperbolically Lipschitz continuity. In this paper we will study the hyperbolically Lipschitz or *bi-lipschitz* continuity of a -harmonic quasiconformal mapping with an angular range and its sharp hyperbolically Lipschitz coefficient determined by the constant of quasiconformality. The main result of this paper is the sharp bounds of their hyperbolically partial derivatives. 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]. The rest of this paper is organized as follows.

In Section 2, using a property of hyperbolic metric of the upper half plane , we first build a differential equation for the hyperbolic metric of an angular domain with the origin of as its vertex (see Lemma 2.1). The two-order differential equation (2.4) is important to derive the upper and lower bounds of the hyperbolically partial derivative of a -harmonic quasiconformal mappings with an angular range.

In Section 3, by combining the well-known Ahlfors-Schwarz lemma and its opposite type given by Mateljević [19] with the differential inequality (2.4), we obtain the upper and lower bounds of the hyperbolically partial derivatives of -harmonic -quasiconformal mappings with angular ranges (see Theorem 3.1). We also show that both the upper and lower bounds of are sharp.

In Section 4, the hyperbolically -*bi-lipschitz* continuity of a -harmonic -quasiconformal mapping with an angular range is obtained by the sharp inequality (3.2) (see Theorem 4.1). The hyperbolically *bi-lipschitz* coefficients are sharp.

At last, some auxiliary examples are given. 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 -harmonic quasiconformal mappings with nonangular ranges (see Examples 5.4 and 5.5).

#### 2. A Differential Equation for the Hyperbolic Metric of an Angular Domain

Let be the hyperbolic metric of the upper half plane with gaussian curvature . Then Hence, the hyperbolic metric of satisfies that By the relation that , the differential equation (2.2) becomes

Using the differential equation (2.3) of the hyperbolic metric of we obtain the following.

Lemma 2.1. *Let be an angular domain with the origin of the complex plane as its vertex. Then for every the hyperbolic metric of satisfies the following differential equation
*

*Proof. *Let be the angular domain with as its vertex and as its hyperbolic metric with gaussian curvature . Let be a conformal mapping of onto . Then by the fact that a hyperbolic metric is a conformal invariant it follows that
Hence by the chain rule [20] we get
From the relations (2.5) and (2.6) we get
Using (2.3) we can simplify the previous relation as
where .

Let . Then is a conformal mapping of onto the upper half plane and the following relations
hold for every . Hence, it follows from the above relations (2.8) and (2.9) that

Let be an arbitrary angular domain only satisfying that its vertex is the origin of . Then there exists a rotation transformation with such that conformally maps onto . Hence,
Thus by the relation (2.10) the following differential equation:
holds for every .

#### 3. Sharp Bounds for Hyperbolically Partial Derivatives

In order to study the hyperbolically *bi-lipschitz* continuity of a -harmonic -quasiconformal mapping, we will first derive the bounds, determined by the quasiconformal constant , of its hyperbolically partial derivative.

To do so we need the well-known Ahlfors-Schwarz lemma [21] and its opposite type given by Mateljević [19] as follows.

Lemma A. *If is a metric density on for which the gaussian curvature satisfies and if tends to when tends to , then .*

Kalaj [7] obtained the following.

Lemma B. *Let be a convex domain in . If is a euclidean harmonic -quasiconformal mapping of the unit disk onto , satisfying , then
**
where and . *

Theorem 3.1. *Let be an angular domain with the origin of the complex plane as its vertex. If is a -harmonic -quasiconformal mapping of the unit disk onto , then for every its hyperbolically partial derivative satisfies the following inequality:
**
Moreover, the upper and lower bound is sharp.*

*Proof. *Let be an angular domain with the origin of the complex plane as its vertex and a -harmonic -quasiconformal mapping of onto . Let . From the assumptions we have that does not vanish on . So is harmonic in . Hence, we have that does not vanish by Lewy Theorem [22]. So also does not vanish. Suppose that , . Therefore for every point . Thus we obtain

By the chain rule [20] we get
By Euler-Lagrange equation we have that a -harmonic mapping satisfies
Since does not vanish, we have from (3.5) that

Using the relations (3.3), (3.4), (3.5), and (3.6) we have
By the differential equation at Lemma 2.1 the above relation becomes
So we get
By (1.2) it is clear that . Hence, it follows from (3.9) and the inequality that

Thus by Ahlfors-Schwarz Lemma [21, P13] it follows that , that is,

Let , . Then is a -harmonic -quasiconformal mapping of onto itself. Moreover, we also have
Choosing to be a conformal mapping of onto , we have that is -harmonic -quasiconformal mapping of onto . Thus by (1.5) the equality (3.12) becomes that
Therefore the upper bound at (3.2) is sharp.

Next we will prove the lower bound of . Suppose that is a -harmonic -quasiconformal mapping of onto . Let .

Hence, we have
Combining Lemma 2.1 with the relations (3.4), (3.5), (3.6), and (3.14) we have
Hence, it follows from the inequality and (3.15) that
Since the mapping maps onto a strip domain , we have that is an euclidean harmonic mapping of onto . So it follows from Lemma B that , where is a positive constant. Thus we have as . Thus it follows from Lemma A that

Let , . Then is a -harmonic -quasiconformal mapping of onto itself. Moreover, we also have
Choosing to be a conformal mapping of onto , we have that is -harmonic -quasiconformal mapping of onto . Thus by (1.5) it shows that
Therefore the positive lower bound at (3.2) is also sharp.

#### 4. Sharp Coefficients of Hyperbolically Lipschitz Continuity

As an application of Theorem 3.1, we have the following main result in this paper.

Theorem 4.1. *Let be an angular domain with the origin of the complex plane as its vertex. If is a -harmonic -quasiconformal mapping of the unit disk onto , then is hyperbolically -bi-lipschitz. Moreover, both the coefficients and are sharp.*

*Proof. *Let be the hyperbolic geodesic between and , where and are two arbitrary points in . Then it follows that
where . By the inequality of (3.2) and the definition of a hyperbolic geodesic, we obtain from the above inequality that
Hence, is hyperbolically -Lipschitz.

Let , . Then is a -harmonic -quasiconformal mapping of onto itself. Let and , be two points in . Then and . Thus and . So the equality
holds. Choosing to be a conformal mapping of onto , we have that is -harmonic -quasiconformal mapping of onto . Let and . Thus by the fact that the hyperbolic distance is a conformal invariant it follows from (1.5) that
Thus the coefficient is sharp.

Let be the hyperbolic geodesic connected with . By the assumption that tends to as , we have that the inequality (3.2) also holds. Hence, we also have
where . Thus is hyperbolically -*bi-lipschitz*.

Let , . Let , and , be two points in . Then and . Thus and . So the equality
holds. Choosing to be a conformal mapping of onto , we have that is -harmonic -quasiconformal mapping of onto . Let and . Thus by the fact that the hyperbolic distance is a conformal invariant it shows that
Thus the coefficient is also sharp. The proof of Theorem 4.1 is complete.

#### 5. Auxiliary Examples

*Example 5.1. *Suppose that . Let be the punctured unit disk and the unit disk. Then is a smooth metric on but not smooth on . We have that is a -harmonic -quasiconformal mapping of onto itself. If a -harmonic mapping is not necessary to be smooth, then is also a -harmonic -quasiconformal mapping of onto itself. Moreover, it follows that

*Example 5.2. *Suppose that . We have that is a -harmonic -quasiconformal mapping of onto itself. If a -harmonic mapping is not necessary to be smooth, then is also a -harmonic -quasiconformal mapping of onto itself. Similar to Example 5.1, it follows that

*Example 5.3. *Suppose that . Then is a -harmonic -quasiconformal mapping of the upper half plane onto itself, here . Moreover,

*Example 5.4. *Let and . Let , . Then is a -harmonic -quasiconformal mapping of onto itself and satisfies that

*Example 5.5. *Let be the right half plane. Let . Then is not an angular domain. The hyperbolic metric with gaussian curvature is given by
Let , , where . Then is a -harmonic -quasiconformal mapping of onto itself. Moreover, we have
where . Let , , where . Then is a -harmonic -quasiconformal mapping of onto itself. Moreover, we have
where .

#### Acknowledgments

Foundation items Supported by NNSF of China (11101165), the Fundamental Research Funds for the Central Universities of Huaqiao university (JB-ZR1136) and NSF of Fujian Province (2011J01011).