Hyperbolically Bi-Lipschitz Continuity for 1/|𝑤|²-Harmonic Quasiconformal Mappings

Abstract

We study the class of 1/|𝑤|²-harmonic 𝐾-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 𝐾. As an application we get their hyperbolically bi-Lipschitz continuity and their sharp hyperbolically bi-Lipschitz coefficients.

1. Introduction

Let Ω and Ω′ be two domains of hyperbolic type in the complex plane ℂ. A 𝐶² sense-preserving homeomorphism 𝑓 of Ω onto Ω′ is said to be a 𝜌-harmonic mapping if it satisfies the Euler-Lagrange equation𝑓_𝑧𝑧+log𝜌_𝑤𝑓𝑓_𝑧𝑓_𝑧=0, (1.1) where 𝑤=𝑓(𝑧) and 𝜌(𝑤)|𝑑𝑤|² is a smooth metric in Ω′. If 𝜌 is a constant then 𝑓 is said to be euclidean harmonic. A euclidean harmonic mapping defined on a simply connected domain is of the form 𝑓=ℎ+𝑔, where ℎ and 𝑔 are two analytic functions in Ω. For a survey of harmonic mappings, see [1–3].

In this paper we study the class of 1/|𝑤|²-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 𝑓_𝑧=𝑎𝑓𝑓𝑓_𝑧, (1.2) where 𝑎(𝑧) is analytic and |𝑎(𝑧)|<1 (see [4–6] for more details). By differentiating (1.2) in 𝑧, we have that 𝑓_𝑧𝑧+1log|𝑤|²_𝑤∘𝑓𝑓_𝑧𝑓_𝑧=𝑎𝑓𝑓𝑓_𝑧𝑧+1log|𝑤|²_𝑤∘𝑓𝑓_𝑧𝑓_𝑧. (1.3) Hence, it follows that a logharmonic mapping is a 1/|𝑤|²-harmonic mapping.

If a 𝜌-harmonic mapping 𝑓 also satisfies the condition that |𝑓_𝑧(𝑧)|≤𝑘|𝑓_𝑧(𝑧)| holds for every 𝑧∈Ω, then it is called a 𝜌-harmonic 𝐾-quasiconformal mapping (for simplicity, a harmonic quasiconformal mapping or H.Q.C mapping), where 𝐾=(1+𝑘)/(1−𝑘).

Let 𝜆_Ω(𝑧)|𝑑𝑧| denote the hyperbolic metric of a simply connected region Ω with gaussian curvature −4. For a harmonic quasiconformal mapping 𝑓 of Ω onto Ω′, we call the quantity =𝜆𝜕𝑓^Ω′∘𝑓𝜆_Ω𝑓_𝑧 (1.4) the hyperbolically partial derivative of 𝑓. If 𝑓 is a harmonic quasiconformal mapping of Ω₁ onto Ω₂ and 𝜑 is a conformal mapping of Ω₀ onto Ω₁ then 𝑓∘𝜑 is also a harmonic quasiconformal mapping. We have 𝜕=𝜆𝑓∘𝜑^Ω2𝑓∘𝜑(𝜁)𝜆_Ω0(𝜁)𝑓∘𝜑_𝜁=𝜆_Ω2𝑓(𝑧)𝜆_Ω1(𝑧)𝑓_𝑧=𝜕𝑓, (1.5) where 𝑧=𝜑(𝜁). Hence, we always fix the domain of a harmonic quasiconformal mapping to be the unit disk 𝐷 when studying its hyperbolically partial derivative.

The hyperbolic distance 𝑑_ℎ(𝑧₁,𝑧₂) between 𝑧₁ and 𝑧₂ is defined by inf_𝛾∫_𝛾𝜆_Ω(𝑧)|𝑑𝑧|, where 𝛾 runs through all rectifiable curves in Ω which connect 𝑧₁ and 𝑧₂. A harmonic quasiconformal mapping 𝑓 of Ω onto Ω′ is said to be hyperbolically 𝐿₁-Lipschitz (𝐿₁>0) if 𝑑_ℎ𝑓𝑧₁,𝑓𝑧₂≤𝐿₁𝑑_ℎ𝑧₁,𝑧₂,𝑧₁,𝑧₂∈Ω. (1.6) The constant 𝐿₁ is said to be the hyperbolically Lipschitz coefficient of 𝑓. If there also exists a constant 𝐿₂>0 such that 𝐿₂𝑑_ℎ𝑧₁,𝑧₂≤𝑑_ℎ𝑓𝑧₁,𝑓𝑧₂,𝑧₁,𝑧₂∈Ω, (1.7) then 𝑓 is said to be hyperbolically (𝐿₂,𝐿₁)-bi-lipschitz. We also call the array (𝐿₂,𝐿₁) the hyperbolically bi-lipschitz coefficient of 𝑓.

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 Ω and Ω′ be two Jordan domains, let 𝛼∈(0,1] and let 𝑓∶Ω↦Ω′ be a euclidean harmonic quasiconformal mapping. If 𝜕Ω and 𝜕Ω′∈𝐶^1,𝛼, then 𝑓 is euclidean Lipschitz. In particular, if Ω′ is convex, then 𝑓 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 𝑓 be a euclidean harmonic 𝐾-quasiconformal mapping of Ω onto a convex domain Ω^′. Then 𝑓 is hyperbolically (1/𝐾,𝐾)-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 𝜌-harmonic quasiconformal mapping also has similar Lipschitz or bi-lipschitz continuity. 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, then 𝑓 generally does not need to have euclidean and hyperbolically Lipschitz continuity even if its range is convex. Hence, we only consider the case that 𝜌 is smooth, that is, 1/|𝑤|² does not vanish in the range of a 1/|𝑤|²-harmonic quasiconformal mapping in this paper. Kalaj and Mateljević (see Theorem 4.4 of [18]) showed the following.

Theorem C. Let 𝜑 be analytic in Ω′ and 𝑓 a |𝜑|-harmonic quasiconformal mapping of the 𝐶^1,𝛼 domain Ω onto the 𝐶^1,𝛼 Jordan domain Ω′. If 𝑀=‖(log𝜑)′‖_∞<∞, then 𝑓 is euclidean Lipschitz.

Let |𝜑(𝑤)| be equal to 1/|𝑤|², where 𝑤∈Ω′. If the closure of the range Ω′ does not include the origin, then 𝑀=‖(log𝜑)′‖_∞=‖1/|𝑤‖|_∞ is finite. So by Theorem C a 1/|𝑤|²-harmonic quasiconformal mapping with such a range Ω′ has euclidean Lipschitz continuity. Example 5.3 shows that if the origin is a boundary point of 𝜕Ω′ then a 1/|𝑤|²-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 1/|𝑤|²-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 ℍ, we first build a differential equation for the hyperbolic metric of an angular domain with the origin of ℂ 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 1/|𝑤|²-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 ‖𝜕𝑓‖ of 1/|𝑤|²-harmonic 𝐾-quasiconformal mappings with angular ranges (see Theorem 3.1). We also show that both the upper and lower bounds of ‖𝜕𝑓‖ are sharp.

In Section 4, the hyperbolically 𝐾-bi-lipschitz continuity of a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping with an angular range is obtained by the sharp inequality (3.2) (see Theorem 4.1). The hyperbolically bi-lipschitz coefficients (1/𝐾,𝐾) 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 1/|𝑤|²-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 𝜆_𝐻(𝑤)|𝑑𝑤| be the hyperbolic metric of the upper half plane ℍ with gaussian curvature −4. Then 𝜆_𝐻(𝑤|=𝑖)|𝑑𝑤𝑤−𝑤|𝑑𝑤|,log𝜆_𝐻𝑤1=−𝑤−𝑤,log𝜆_𝐻𝑤𝑤=1(𝑤−𝑤)². (2.1) Hence, the hyperbolic metric 𝜆_𝐻(𝑤)|𝑑𝑤| of ℍ satisfies that log𝜆_𝐻𝑤𝑤+log𝜆_𝐻𝑤𝑤+𝑤𝑤𝜆²_𝐻=0. (2.2) By the relation that (log𝜆_𝐻)_𝑤=(𝜆_𝐻)_𝑤/𝜆_𝐻, the differential equation (2.2) becomes 𝜆_𝐻𝑤𝑤𝜆_𝐻=𝜆_𝐻𝑤𝜆_𝐻²−𝜆_𝐻𝑤𝑤𝜆_𝐻−𝑤𝑤𝜆_𝐻². (2.3)

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

Lemma 2.1. Let 𝐴 be an angular domain with the origin of the complex plane ℂ as its vertex. Then for every 𝜁∈𝐴 the hyperbolic metric 𝜆_𝐴(𝜁)|𝑑𝜁| of 𝐴 satisfies the following differential equation log𝜆_𝐴𝜁𝜁+log𝜆_𝐴𝜁𝜁+𝜁𝜁𝜆_𝐴²=0. (2.4)

Proof. Let 𝐴_𝜃 be the angular domain {𝑧∈ℂ∣0<arg𝑧<𝜃,𝜃∈(0,2𝜋]} with 0 as its vertex and 𝜆_𝐴𝜃(𝑧)|𝑑𝑧| as its hyperbolic metric with gaussian curvature −4. Let 𝑓 be a conformal mapping of 𝐴_𝜃 onto ℍ. Then by the fact that a hyperbolic metric is a conformal invariant it follows that 𝜆_𝐴𝜃(𝑧)=𝜆_𝐻∘𝑓𝑓^′. (2.5) Hence by the chain rule [20] we get log𝜆_𝐴𝜃𝑧=𝜆_𝐻𝑤∘𝑓𝑓′𝜆_𝐻+1∘𝑓2𝑓^′′,𝑓′log𝜆_{𝐴𝜃𝑧𝑧}=𝜆_𝐻∘𝑓𝜆_𝐻𝑤𝑤∘𝑓𝑓^′2+𝜆_𝐻𝑤∘𝑓𝑓^′′−𝜆_𝐻𝑤∘𝑓𝑓^′2𝜆_𝐻∘𝑓²+𝑓^′′2𝑓^′′. (2.6) From the relations (2.5) and (2.6) we get log𝜆_{𝐴𝜃𝑧𝑧}+log𝜆_𝐴𝜃𝑧𝑧+𝑧𝑧𝜆_𝐴𝜃²=𝜆_𝐻𝑤𝑤𝜆_𝐻∘𝑓𝑓^′2+𝜆_𝐻𝑤𝜆_𝐻∘𝑓𝑓^′′−𝜆_𝐻𝑤𝜆_𝐻∘𝑓²𝑓^′2+12𝑓^′′𝑓^′′+𝜆_𝐻𝑤𝜆_𝐻𝑓∘𝑓^′𝑧+1𝑓2𝑧𝑓^′′′+𝑧𝑧𝜆_𝐻∘𝑓𝑓^′2. (2.7) Using (2.3) we can simplify the previous relation aslog𝜆_{𝐴𝜃𝑧𝑧}+log𝜆_𝐴𝜃𝑧𝑧+𝑧𝑧𝜆_𝐴𝜃²=𝜆_𝐻𝑤𝜆_𝐻∘𝑓𝑓^′′+𝑓^′𝑧−𝑓^′2𝑓+12𝑓^′′𝑓^′′+1𝑓2𝑧𝑓^′′′+𝜆_𝐻∘𝑓²𝑧𝑧𝑓^′2−𝑓𝑓𝑓^′2, (2.8) where 𝑤=𝑓(𝑧).
Let 𝑓(𝑧)=𝑧^𝛼,𝛼∈[1/2,1)∪(1,∞). Then 𝑓 is a conformal mapping of 𝐴_𝜃 onto the upper half plane ℍ and the following relations 𝑓^′′+𝑓^′𝑧−𝑓^′2𝑓1=0,2𝑓^′′𝑓^′′+1𝑓2𝑧𝑓^′′′=0,𝑧𝑧𝑓^′2−𝑓𝑓𝑓′²=0 (2.9) hold for every 𝑧∈𝐴_𝜃. Hence, it follows from the above relations (2.8) and (2.9) that log𝜆_{𝐴𝜃𝑧𝑧}+log𝜆_𝐴𝜃𝑧𝑧+𝑧𝑧𝜆_𝐴𝜃²=0. (2.10)
Let 𝐴 be an arbitrary angular domain only satisfying that its vertex is the origin of ℂ. Then there exists a rotation transformation 𝑧=𝑔(𝜁)=𝑒^𝑖𝜃0𝜁,𝜁∈𝐴 with 0≤𝜃₀≤2𝜋 such that 𝑔 conformally maps 𝐴 onto 𝐴_𝜃. Hence, 𝜆_𝐴(𝜁)=𝜆_𝐴𝜃𝑔(𝜁),log𝜆_𝐴(𝜁)_𝜁=𝑒^𝑖𝜃0log𝜆_𝜃(𝑧)_𝑧,log𝜆_𝐴(𝜁)_𝜁𝜁=𝑒^2𝑖𝜃0log𝜆_𝜃(𝑧)_𝑧𝑧. (2.11) Thus by the relation (2.10) the following differential equation: log𝜆_𝐴𝜁𝜁+log𝜆_𝐴𝜁𝜁+𝜁𝜁𝜆_𝐴²=0 (2.12) holds for every 𝜁∈𝐴.

3. Sharp Bounds for Hyperbolically Partial Derivatives

In order to study the hyperbolically bi-lipschitz continuity of a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping, we will first derive the bounds, determined by the quasiconformal constant 𝐾, 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 𝜌>0 is a 𝐶² metric density on 𝐷 for which the gaussian curvature satisfies 𝐾_𝜌≥−4 and if 𝜌(𝑧) tends to +∞ when |𝑧| tends to 1⁻, then 𝜆_𝐷≤𝜌.

Kalaj [7] obtained the following.

Lemma B. Let Ω be a convex domain in 𝐶. If 𝑓 is a euclidean harmonic 𝐾-quasiconformal mapping of the unit disk onto Ω, satisfying 𝑓(0)=𝑎, then 𝑓_𝑧≥12()𝛿1+𝑘^Ω,𝑧∈𝐷, (3.1) where 𝛿_Ω=𝑑(𝑎,𝜕Ω)=inf{|𝑓−𝑎|∶𝑓∈𝜕Ω} and 𝑘=(𝐾−1)/(𝐾+1).

Theorem 3.1. Let 𝐴 be an angular domain with the origin of the complex plane ℂ as its vertex. If 𝑓 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of the unit disk 𝐷 onto 𝐴, then for every 𝑧∈𝐷 its hyperbolically partial derivative satisfies the following inequality: 𝐾+1≤≤2𝐾𝜕𝑓𝐾+12. (3.2) Moreover, the upper and lower bound is sharp.

Proof. Let 𝐴 be an angular domain with the origin of the complex plane ℂ as its vertex and 𝑓 a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto 𝐴. Let 𝑘=(𝐾−1)/(𝐾+1). From the assumptions we have that 𝑓 does not vanish on 𝐷. So log𝑓 is harmonic in Ω. Hence, we have that (log𝑓)_𝑧 does not vanish by Lewy Theorem [22]. So 𝑓_𝑧 also does not vanish. Suppose that 𝜎(𝑧)=(1−𝑘)𝜆_𝐴(𝑓(𝑧))|𝑓_𝑧|, 𝑧∈𝐷. Therefore 𝜎(𝑧)>0 for every point 𝑧∈𝐷. Thus we obtain Δlog𝜎(𝑧)=4log𝜆_𝐴∘𝑓_𝑧𝑧(𝑧)+log𝑓_𝑧𝑧𝑧. (3.3)
By the chain rule [20] we get 4log𝜆_𝐴∘𝑓_𝑧𝑧(𝑧)=4log𝜆_𝐴𝑤𝑤∘𝑓𝑓_𝑧²+𝑓_𝑧²+2ℜlog𝜆_𝐴𝑤𝑤∘𝑓𝑓_𝑧𝑓_𝑧+2ℜlog𝜆_𝐴𝑤∘𝑓𝑓_𝑧𝑧. (3.4) By Euler-Lagrange equation we have that a 1/|𝑤|²-harmonic mapping 𝑓 satisfies 𝑓_𝑧𝑧−𝑓_𝑧𝑓_𝑧𝑓=0. (3.5) Since 𝑓_𝑧 does not vanish, we have from (3.5) that log𝑓_𝑧𝑧𝑧=0. (3.6)
Using the relations (3.3), (3.4), (3.5), and (3.6) we have Δlog𝜎(𝑧)=4log𝜆_𝐴𝑤𝑤∘𝑓𝑓_𝑧²+𝑓_𝑧²+2ℜlog𝜆_𝐴𝑤𝑤+log𝜆_𝐴𝑤𝑤∘𝑓𝑓_𝑧𝑓_𝑧. (3.7) By the differential equation at Lemma 2.1 the above relation becomes Δlog𝜎(𝑧)=4log𝜆_𝐴𝑤𝑤∘𝑓𝑓_𝑧²+𝑓_𝑧²−2ℜ𝜆_𝐴∘𝑓²𝑓𝑓_𝑧𝑓_𝑧𝑓. (3.8) So we get −Δlog𝜎𝜎²=−4(1−𝑘)²Δlog𝜆_𝐴4𝜆_𝐴²𝑓∘𝑓^𝑧2+𝑓_𝑧²𝑓_𝑧²−2ℜ𝑓𝑓_𝑧𝑓𝑓_𝑧. (3.9) By (1.2) it is clear that |𝑓_𝑧/𝑓_𝑧|=|𝑎|. Hence, it follows from (3.9) and the inequality |𝑎|≤𝑘 that 𝐾_𝜎=−Δlog𝜎𝜎²4≤−(1−𝑘)²1+|𝑎|²|𝑎|(|𝑎−2=−41−|)²(1−𝑘)²≤−4. (3.10)
Thus by Ahlfors-Schwarz Lemma [21, P13] it follows that 𝜎≤𝜆_𝐷, that is,=𝜆𝜕𝑓^𝐴∘𝑓𝜆_𝐷𝑓_𝑧≤𝐾+12. (3.11)
Let 𝐹=𝑤|𝑤|^𝐾−1, 𝑤∈ℍ. Then 𝐹 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of ℍ onto itself. Moreover, we also have ‖‖=𝜆𝜕𝐹^𝐻∘𝐹𝜆_𝐻𝐹_𝑤=𝐾+12. (3.12) Choosing 𝐿 to be a conformal mapping of 𝐷 onto ℍ, we have that 𝐹∘𝐿 is 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto ℍ. Thus by (1.5) the equality (3.12) becomes that ‖𝜕(𝐹∘𝐿)‖=𝐾+12. (3.13) Therefore the upper bound at (3.2) is sharp.
Next we will prove the lower bound of ‖𝜕𝑓‖. Suppose that 𝑓 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto 𝐴. Let 𝛿=(1+𝑘)𝜆_𝐴(𝑓)|𝑓_𝑧|.
Hence, we have Δlog𝛿(𝑧)=4log𝜆_𝐴∘𝑓_𝑧𝑧(𝑧)+log𝑓_𝑧𝑧𝑧. (3.14) Combining Lemma 2.1 with the relations (3.4), (3.5), (3.6), and (3.14) we have −Δlog𝛿𝛿²=−4(1+𝑘)²Δlog𝜆_𝐴4𝜆_𝐴²𝑓∘𝑓^𝑧2+𝑓_𝑧²𝑓_𝑧²−2ℜ𝑓𝑓_𝑧𝑓𝑓_𝑧. (3.15) Hence, it follows from the inequality |𝑎|≤𝑘 and (3.15) that 𝐾_𝛿=−Δlog𝛿𝛿²4≥−(1+𝑘)²1+|𝑎|²|𝑎|(|𝑎+2=−41+|)²(1+𝑘)²≥−4. (3.16) Since the mapping log𝑤 maps 𝐴 onto a strip domain 𝑆, we have that log𝑓 is an euclidean harmonic mapping of 𝐷 onto 𝑆. So it follows from Lemma B that |(log𝑓)_𝑧|≥𝐶₀, where 𝐶₀ is a positive constant. Thus we have 𝜆_𝐴(𝑓)|𝑓_𝑧|=𝜆_𝑆(log𝑓)|(log𝑓)_𝑧|→+∞ as |𝑧|→1⁻. Thus it follows from Lemma A that =𝜆𝜕𝑓^𝐴∘𝑓𝜆_𝐷𝑓_𝑧≥𝐾+1.2𝐾 (3.17)
Let 𝐹=𝑤|𝑤|^1/𝐾−1, 𝑤∈ℍ. Then 𝐹 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of ℍ onto itself. Moreover, we also have ‖‖=𝜆𝜕𝐹^𝐻∘𝐹𝜆_𝐻𝐹_𝑤=𝐾+1.2𝐾 (3.18) Choosing 𝐿 to be a conformal mapping of 𝐷 onto ℍ, we have that 𝐹∘𝐿 is 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto ℍ. Thus by (1.5) it shows that ‖𝜕(𝐹∘𝐿)‖=𝐾+1.2𝐾 (3.19) 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 𝐴 be an angular domain with the origin of the complex plane ℂ as its vertex. If 𝑓 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of the unit disk 𝐷 onto 𝐴, then 𝑓 is hyperbolically (1/𝐾,𝐾)-bi-lipschitz. Moreover, both the coefficients 𝐾 and 1/𝐾 are sharp.

Proof. Let 𝛾 be the hyperbolic geodesic between 𝑧₁ and 𝑧₂, where 𝑧₁ and 𝑧₂ are two arbitrary points in 𝐷. Then it follows that _𝑓(𝛾)𝜆_𝐴(𝑤)|𝑑𝑤|≤_𝛾𝜆_𝐴𝑓(𝑧)𝐿_𝑓(𝑧)|𝑑𝑧|≤2𝐾𝐾+1_𝛾𝜆_𝐴𝑓(𝑧)𝑓_𝑧(𝑧)𝜆_𝐷(𝑧)𝜆_𝐷(𝑧)|𝑑𝑧|, (4.1) where 𝑤=𝑓(𝑧). By the inequality of (3.2) and the definition of a hyperbolic geodesic, we obtain from the above inequality that 𝑑_ℎ𝑓𝑧₁,𝑓𝑧₂≤_𝑓(𝛾)𝜆_𝐴(𝑤)|𝑑𝑤|≤𝐾_𝛾𝜆_𝐷(𝑧)|𝑑𝑧|=𝐾𝑑_ℎ𝑧₁,𝑧₂. (4.2) Hence, 𝑓 is hyperbolically 𝐾-Lipschitz.
Let 𝐹=𝑤|𝑤|^𝐾−1, 𝑤∈ℍ. Then 𝐹 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of ℍ onto itself. Let 𝑧₁=𝑖 and 𝑧₂=𝑖𝑦, 𝑦>1 be two points in ℍ. Then 𝐹(𝑧₁)=𝑖 and 𝐹(𝑧₂)=𝑖𝑦^𝐾. Thus 𝑑_ℎ(𝑧₁,𝑧₂)=log𝑦 and 𝑑_ℎ(𝐹(𝑧₁),𝐹(𝑧₂))=𝐾log𝑦. So the equality 𝑑_ℎ𝐹𝑧₁,𝐹𝑧₂=𝐾𝑑_ℎ𝑧₁,𝑧₂ (4.3) holds. Choosing 𝐿 to be a conformal mapping of 𝐷 onto ℍ, we have that 𝜙=𝐹∘𝐿 is 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto ℍ. Let 𝜙(𝜁₁)=𝑧₁ and 𝜙(𝜁₂)=𝑧₂. Thus by the fact that the hyperbolic distance is a conformal invariant it follows from (1.5) that𝑑_ℎ𝜙𝜁₁,𝜙𝜁₂=𝐾𝑑_ℎ𝐿𝜁₁,𝐿𝜁₂=𝐾𝑑_ℎ𝜁₁,𝜁₂. (4.4) Thus the coefficient 𝐾 is sharp.
Let 𝑓(𝛾)⊂𝐴 be the hyperbolic geodesic connected 𝑓(𝑧₁) with 𝑓(𝑧₂). By the assumption that 𝜆_𝐴|𝑓_𝑧| tends to +∞ as |𝑧|→1⁻, we have that the inequality (3.2) also holds. Hence, we also have𝑑_ℎ𝑓𝑧₁,𝑓𝑧₂=_𝑓(𝛾)𝜆_𝐴(𝑤|≥1)|𝑑𝑤𝐾_𝛾𝜆_𝐷(𝑧|≥1)|𝑑𝑧𝐾𝑑_ℎ𝑧₁,𝑧₂, (4.5) where 𝑤=𝑓(𝑧). Thus 𝑓 is hyperbolically (1/𝐾,𝐾)-bi-lipschitz.
Let 𝐺=𝑤|𝑤|^1/𝐾−1, 𝑤∈ℍ. Let 𝑧₁=𝑖, and 𝑧₂=𝑖𝑦, 𝑦>1 be two points in ℍ. Then 𝐺(𝑧₁)=𝑖 and 𝐺(𝑧₂)=𝑖𝑦^1/𝐾. Thus 𝑑_ℎ(𝑧₁,𝑧₂)=log𝑦 and 𝑑_ℎ(𝐺(𝑧₁),𝐺(𝑧₂))=(1/𝐾)log𝑦. So the equality𝑑_ℎ𝐺𝑧₁,𝐺𝑧₂=𝑑_ℎ𝑧₁,𝑧₂𝐾 (4.6) holds. Choosing 𝐿 to be a conformal mapping of 𝐷 onto ℍ, we have that 𝜓=𝐺∘𝐿 is 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto ℍ. Let 𝜓(𝜁₁)=𝑧₁ and 𝜓(𝜁₂)=𝑧₂. Thus by the fact that the hyperbolic distance is a conformal invariant it shows that𝑑_ℎ𝜓𝜁₁,𝜓𝜁₂=𝑑_ℎ𝐿𝜁₁,𝐿𝜁₂𝐾=𝑑_ℎ𝜁₁,𝜁₂𝐾. (4.7) Thus the coefficient 1/𝐾 is also sharp. The proof of Theorem 4.1 is complete.

5. Auxiliary Examples

Example 5.1. Suppose that 𝑓=𝑧|𝑧|^1/𝐾−1,𝐾>1. Let 𝐷^∗={𝑧∣0<|𝑧|<1} be the punctured unit disk and 𝐷={𝑧∣|𝑧|<1} the unit disk. Then 1/|𝑤|² is a smooth metric on 𝐷^∗ but not smooth on 𝐷. We have that 𝑓 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷^∗ onto itself. If a 𝜌-harmonic mapping is not necessary to be smooth, then 𝑓 is also a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto itself. Moreover, it follows that lim_𝑧→0𝜆_𝐷𝑓(𝑧)𝜆_𝐷(𝑧)𝑓_𝑧=lim_𝑧→01−𝑟²1−𝑟^2/𝐾(1/𝐾+1)|𝑧|^1/𝐾−12=∞,lim_𝑧→0𝑓(𝑧)−𝑓(0)|𝑧−0|=lim_𝑧→0|𝑧|^1/𝐾−1=∞,lim_𝑧→0𝑑_ℎ𝑓(0),𝑓(𝑧)𝑑_ℎ(0,𝑧)=lim_𝑧→0log1+|𝑧|^1/𝐾/1−|𝑧|^1/𝐾log((1+|𝑧|)/(1−|𝑧|))=lim_𝑧→0|𝑧|^1/𝐾−1=∞,lim_𝑧→0𝜆_𝐷∗𝑓(𝑧)𝜆_𝐷∗(𝑧)𝑓_𝑧=lim_𝑧→0|𝑧|log(1/|𝑧|)|𝑧|^1/𝐾log1/|𝑧|^1/𝑘(1/𝐾+1)|𝑧|^1/𝑘−12=𝐾+12. (5.1)

Example 5.2. Suppose that 𝑓=𝑧|𝑧|^𝐾−1,𝐾>1. We have that 𝑓 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷^∗ onto itself. If a 𝜌-harmonic mapping is not necessary to be smooth, then 𝑓 is also a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of 𝐷 onto itself. Similar to Example 5.1, it follows thatlim_𝑧→0𝜆_𝐷𝑓(𝑧)𝜆_𝐷(𝑧)𝑓_𝑧=lim_𝑧→0𝐾+121−𝑟²1−𝑟^2𝐾𝑟^𝐾−1=0,lim_𝑧→0|𝑧−0|𝑓(𝑧)−𝑓(0)=lim_𝑧→0|𝑧|^1−𝐾=∞,lim_𝑧→0𝑑_ℎ(0,𝑧)𝑑_ℎ𝑓(0),𝑓(𝑧)=lim_𝑧→0log((1+|𝑧|)/(1−|𝑧|))log1+|𝑧|^𝐾/1−|𝑧|^𝐾=lim_𝑧→0|𝑧|^1−𝐾=∞,lim_𝑧→0𝜆_𝐷∗𝑓(𝑧)𝜆_𝐷∗(𝑧)𝑓_𝑧=lim_𝑧→0|𝑧|log(1/|𝑧|)|𝑧|^𝐾log1/|𝑧|^𝐾𝐾+12|𝑧|^𝐾−1=𝐾+1.2𝐾 (5.2)

Example 5.3. Suppose that 𝑓(𝑧)=𝑧|𝑧|^𝐾−1,𝐾>1. Then 𝑓 is a |𝜑|-harmonic 𝐾-quasiconformal mapping of the upper half plane 𝐻 onto itself, here 𝜑(𝑤)=1/𝑤². Moreover, lim_|𝑧|→∞𝑓_𝑧=lim_|𝑧|→∞𝐾+1|𝑧|2𝐾^𝐾−1=+∞,log𝜑(𝑤)_𝑤=𝜑_𝑤𝜑=1𝑤⟶∞,𝑤⟶0,lim_𝑧→∞𝑓(𝑧)|𝑧|=lim_𝑧→∞|𝑧|^𝐾−1=∞,𝜕𝑓=𝐾+12. (5.3)

Example 5.4. Let Ω^∗=ℂ⧵𝐷⋃{∞} and 𝐾>1. Let 𝜑(𝑤)=1/𝑤², 𝑤∈Ω^∗. Then 𝑓=𝑧|𝑧|^1/𝐾−1 is a |𝜑|-harmonic 𝐾-quasiconformal mapping of Ω^∗ onto itself and satisfies that log𝜑(𝑤)_𝑤=𝜑_𝑤𝜑=1𝑤≤1,lim_𝑧→0𝜕𝑓=lim_𝑧→∞𝑟²−1𝑟^2/𝐾−1(1/𝐾+1)|𝑧|^1/𝐾−12=∞,lim_𝑟→∞log1+1/𝑟^1/𝐾/1−1/𝑟^1/𝐾log((1+1/𝑟)/(1−1/𝑟))=∞. (5.4)

Example 5.5. Let 𝕌⁺ be the right half plane. Let Ω=𝕌⁺⧵[1,+∞). Then Ω is not an angular domain. The hyperbolic metric 𝜆_Ω(𝑧)|𝑑𝑧| with gaussian curvature −4 is given by 𝜆_Ω(𝑧|=1)|𝑑𝑧√𝑧²+1+√𝑧²𝑧+1√𝑧²|.+1|𝑑𝑧 (5.5) Let 𝑓(𝑧)=𝑧|𝑧|^1/𝐾−1, 𝑧∈Ω, where 𝐾>1. Then 𝑓 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of Ω onto itself. Moreover, we have lim_𝑧→0𝜕𝑓=lim_𝑧→0𝐾+1√2𝐾𝑧²+1√𝑤²√+1𝑧²+1+√𝑧²+1√𝑤²+1+√𝑤²|𝑧|+1^{2(1/𝐾−1)}=∞, (5.6) where 𝑤=𝑓(𝑧). Let 𝑔(𝑧)=𝑧|𝑧|^𝐾−1, 𝑧∈Ω, where 𝐾>1. Then 𝑔 is a 1/|𝑤|²-harmonic 𝐾-quasiconformal mapping of Ω onto itself. Moreover, we have lim_𝑧→0𝜕𝑔=lim_𝑧→0𝐾+12√𝑧²+1𝜉²√+1𝑧²+1+√𝑧²+1𝜉²+1+𝜉²|𝑧|+1^2(𝐾−1)=0, (5.7) where 𝜉=𝑔(𝑧).

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).