# Hyperbolically Bi-Lipschitz Continuity for1/|w|2-Harmonic Quasiconformal Mappings

Authors: Xingdi Chen
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We study the class of1/|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.


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## Body

## 1. Introduction

LetΩ 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 equation(1.1)fzz¯+(log⁡⁡ρ)w(f)fzfz¯=0,
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 of1/|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
(1.2)fz¯¯=(af¯f)fz,
where a(z) is analytic and |a(z)|<1 (see [4–6] for more details). By differentiating (1.2) in z¯, we have that
(1.3)fzz¯+(log⁡1|w|2)w∘ffzfz¯¯=af¯f(fzz¯+(log⁡⁡1|w|2)w∘ffzfz¯).
Hence, it follows that a logharmonic mapping is a 1/|w|2-harmonic mapping.If aρ-harmonic mapping f also satisfies the condition that |fz¯(z)|≤k|fz(z)| holds for every z∈Ω, then it is called a ρ-harmonicK-quasiconformal mapping (for simplicity, a harmonic quasiconformal mapping or H.Q.C mapping), where K=(1+k)/(1-k).LetλΩ(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
(1.4)‖∂f‖=λΩ′∘fλΩ|fz|
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
(1.5)‖∂(f∘φ)‖=λΩ2(f∘φ(ζ))λΩ0(ζ)|(f∘φ)ζ|=λΩ2(f(z))λΩ1(z)|fz|=‖∂f‖,
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 distancedh(z1,z2) between z1 and z2 is defined by infγ∫γλΩ(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 hyperbolicallyL1-Lipschitz(L1>0) if
(1.6)dh(f(z1),f(z2))≤L1dh(z1,z2),z1,z2∈Ω.
The constant L1 is said to be the hyperbolically Lipschitz coefficient of f. If there also exists a constant L2>0 such that
(1.7)L2dh(z1,z2)≤dh(f(z1),f(z2)),z1,z2∈Ω,
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.
LetΩ 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′∈C1,α, then f is euclidean Lipschitz. In particular, if Ω′ is convex, then f is euclidean bi-lipschitz.Recently, the hyperbolically Lipschitz orbi-lipschitz continuity of euclidean harmonic quasiconformal mappings also excited much interest (see [14–17]). In [14], Chen and Fang proved the following.Theorem B.
Letf 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 hyperbolicallybi-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, Examples5.1, 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.
Letφ be analytic in Ω′ and f a |φ|-harmonic quasiconformal mapping of the C1,α domain          Ω  onto the C1,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 Section2, 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 Section3, 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 Section4, 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 Theorems3.1 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 ofan Angular Domain

LetλH(w)|dw| be the hyperbolic metric of the upper half plane ℍ with gaussian curvature -4. Then
(2.1)λH(w)|dw|=iw-w¯|dw|,(log⁡λH)w=-1w-w¯,(log⁡λH)ww=1(w-w¯)2.
Hence, the hyperbolic metric λH(w)|dw| of ℍ satisfies that
(2.2)(log⁡λH)ww+(log⁡λH)ww+w¯wλH2=0.
By the relation that (logλH)w=(λH)w/λH, the differential equation (2.2) becomes
(2.3)(λH)wwλH=((λH)wλH)2-(λH)wwλH-w¯w(λH)2.Using the differential equation (2.3) of the hyperbolic metric of ℍ we obtain the following.Lemma 2.1.
LetA be an angular domain with the origin of the complex plane ℂ as its vertex. Then for every ζ∈A the hyperbolic metric λA(ζ)|dζ| of A satisfies the following differential equation
(2.4)(log⁡λA)ζζ+(log⁡λA)ζζ+ζ¯ζ(λA)2=0.Proof.
LetAθ be the angular domain {z∈ℂ∣0<argz<θ,θ∈(0,2π]} with 0 as its vertex and λAθ(z)|dz| as its hyperbolic metric with gaussian curvature -4. Let f be a conformal mapping of Aθ onto ℍ. Then by the fact that a hyperbolic metric is a conformal invariant it follows that
(2.5)λAθ(z)=λH∘f|f′|.
Hence by the chain rule [20] we get
(2.6)(log⁡λAθ)z=(λH)w∘ff′λH∘f+12f′′f′,(log⁡λAθ)zz=λH∘f[(λH)ww∘ff′2+(λH)w∘ff′′]-[(λH)w∘ff′]2(λH∘f)2+(f′′2f′)′.
From the relations (2.5) and (2.6) we get
(2.7)(log⁡λAθ)zz+(log⁡λAθ)zz+z¯z(λAθ)2=(λH)wwλH∘ff′2+(λH)wλH∘ff′′-[(λH)wλH∘f]2f′2+12(f′′f′)′+(λH)wλH∘ff′z+12zf′′f′+z¯z(λH∘f|f′|)2.
Using (2.3) we can simplify the previous relation as(2.8)(log⁡λAθ)zz+(log⁡λAθ)zz+z¯z(λAθ)2(log⁡λAθ)=(λH)wλH∘f(f′′+f′z-f′2f)+12(f′′f′)′+12zf′′f′+(λH∘f)2(z¯z|f′|2-f¯ff′2),
where w=f(z).
Letf(z)=zα,α∈[1/2,1)∪(1,∞). Then f is a conformal mapping of Aθ onto the upper half plane ℍ and the following relations
(2.9)f′′+f′z-f′2f=0,12(f′′f′)′+12zf′′f′=0,z¯z|f′|2-f¯ff′2=0
hold for every z∈Aθ. Hence, it follows from the above relations (2.8) and (2.9) that
(2.10)(log⁡λAθ)zz+(log⁡λAθ)zz+z¯z(λAθ)2=0.
LetA be an arbitrary angular domain only satisfying that its vertex is the origin of ℂ. Then there exists a rotation transformation z=g(ζ)=eiθ0ζ,ζ∈A with 0≤θ0≤2π such that g conformally maps A onto Aθ. Hence,
(2.11)λA(ζ)=λAθ(g(ζ)),(log⁡λA(ζ))ζ=eiθ0(log⁡λθ(z))z,(log⁡λA(ζ))ζζ=e2iθ0(log⁡λθ(z))zz.
Thus by the relation (2.10) the following differential equation:
(2.12)(log⁡λA)ζζ+(log⁡λA)ζζ+ζ¯ζ(λA)2=0
holds for every ζ∈A.


## 3. Sharp Bounds for Hyperbolically Partial Derivatives

In order to study the hyperbolicallybi-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.
Ifρ>0 is a C2 metric density on D for which the gaussian curvature satisfies Kρ≥-4 and if ρ(z) tends to +∞ when |z| tends to 1-, then λD≤ρ.Kalaj [7] obtained the following.Lemma B.
LetΩ 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
(3.1)|fz|≥12(1+k)δΩ,z∈D,
where δΩ=d(a,∂Ω)=inf{|f-a|:f∈∂Ω} and k=(K-1)/(K+1).Theorem 3.1.
LetA 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:
(3.2)K+12K≤‖∂f‖≤K+12.
Moreover, the upper and lower bound is sharp.Proof.
LetA 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 fz also does not vanish. Suppose that σ(z)=(1-k)λA(f(z))|fz|, z∈D. Therefore σ(z)>0 for every point z∈D. Thus we obtain
(3.3)(Δlog⁡σ)(z)=4[(log⁡λA∘f)zz¯(z)+(log⁡|fz|)zz¯].
By the chain rule [20] we get
(3.4)4(log⁡⁡λA∘f)zz¯(z)=4{((log⁡⁡λA)ww¯∘f)(|fz|2+|fz¯|2)2R+2R[((log⁡⁡λA)ww∘f)fzfz¯]+2R[(log⁡⁡λA)w∘ffzz¯]{((log⁡⁡λA)ww¯∘f)(|fz|2+|fz¯|2)}}.
By Euler-Lagrange equation we have that a 1/|w|2-harmonic mapping f satisfies
(3.5)fzz¯-fzfz¯f=0.
Since fz does not vanish, we have from (3.5) that
(3.6)(log⁡|fz|)zz¯=0.
Using the relations (3.3), (3.4), (3.5), and (3.6) we have
(3.7)(Δlog⁡σ)(z)=4{(log⁡λA)ww¯∘f(|fz|2+|fz¯|2)+2R[[(log⁡λA)ww+(log⁡λA)ww]∘ffzfz¯]}.
By the differential equation at Lemma 2.1 the above relation becomes
(3.8)(Δlog⁡σ)(z)=4{(log⁡λA)ww¯∘f(|fz|2+|fz¯|2)-2R[(λA∘f)2f¯fzfz¯f]}.
So we get
(3.9)-Δlog⁡σσ2=-4(1-k)2[Δlog⁡λA4(λA)2∘f|fz|2+|fz¯|2|fz|2-2Rf¯fz¯ffz¯].
By (1.2) it is clear that |fz¯/fz|=|a|. Hence, it follows from (3.9) and the inequality |a|≤k that
(3.10)Kσ=-Δlog⁡σσ2≤-4(1-k)2(1+|a|2-2|a|)=-4(1-|a|)2(1-k)2≤-4.
Thus by Ahlfors-Schwarz Lemma [21, P13] it follows that σ≤λD, that is,(3.11)‖∂f‖=λA∘fλD|fz|≤K+12.
LetF=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
(3.12)‖∂F‖=λH∘FλH|Fw|=K+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
(3.13)‖∂(F∘L)‖=K+12.
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)λA(f)|fz|.
Hence, we have(3.14)(Δlog⁡δ)(z)=4[(log⁡λA∘f)zz¯(z)+(log⁡|fz|)zz¯].
Combining Lemma 2.1 with the relations (3.4), (3.5), (3.6), and (3.14) we have
(3.15)-Δlog⁡δδ2=-4(1+k)2[Δlog⁡λA4(λA)2∘f|fz|2+|fz¯|2|fz|2-2Rf¯fz¯ffz¯].
Hence, it follows from the inequality |a|≤k and (3.15) that
(3.16)Kδ=-Δlog⁡δδ2≥-4(1+k)2(1+|a|2+2|a|)=-4(1+|a|)2(1+k)2≥-4.
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 λA(f)|fz|=λS(logf)|(logf)z|→+∞ as |z|→1-. Thus it follows from Lemma A that
(3.17)‖∂f‖=λA∘fλD|fz|≥K+12K.
LetF=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
(3.18)‖∂F‖=λH∘FλH|Fw|=K+12K.
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
(3.19)‖∂(F∘L)‖=K+12K.
Therefore the positive lower bound at (3.2) is also sharp.


## 4. Sharp Coefficients of Hyperbolically Lipschitz Continuity

As an application of Theorem3.1, we have the following main result in this paper.Theorem 4.1.
LetA 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.
Letγ be the hyperbolic geodesic between z1 and z2, where z1 and z2 are two arbitrary points in D. Then it follows that
(4.1)∫f(γ)‍λA(w)|dw|≤∫γ‍λA(f(z))Lf(z)|dz|≤2KK+1∫γ‍λA(f(z))|fz(z)|λD(z)λD(z)|dz|,
where w=f(z). By the inequality of (3.2) and the definition of a hyperbolic geodesic, we obtain from the above inequality that
(4.2)dh(f(z1),f(z2))≤∫f(γ)‍λA(w)|dw|≤K∫γ‍λD(z)|dz|=Kdh(z1,z2).
Hence, f is hyperbolically K-Lipschitz.
LetF=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)=iyK. Thus dh(z1,z2)=logy  and dh(F(z1),F(z2))=Klogy. So the equality
(4.3)dh(F(z1),F(z2))=Kdh(z1,z2)
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) that(4.4)dh(ϕ(ζ1),ϕ(ζ2))=Kdh(L(ζ1),L(ζ2))=Kdh(ζ1,ζ2).
Thus the coefficient K is sharp.
Letf(γ)⊂A be the hyperbolic geodesic connected f(z1) with f(z2). By the assumption that λA|fz| tends to +∞ as |z|→1-, we have that the inequality (3.2) also holds. Hence, we also have(4.5)dh(f(z1),f(z2))=∫f(γ)‍λA(w)|dw|≥1K∫γ‍λD(z)|dz|≥1Kdh(z1,z2),
where w=f(z). Thus f is hyperbolically (1/K,K)-bi-lipschitz.
LetG=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)=iy1/K. Thus dh(z1,z2)=logy and dh(G(z1),G(z2))=(1/K)logy. So the equality(4.6)dh(G(z1),G(z2))=dh(z1,z2)K
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 that(4.7)dh(ψ(ζ1),ψ(ζ2))=dh(L(ζ1),L(ζ2))K=dh(ζ1,ζ2)K.
Thus the coefficient 1/K is also sharp. The proof of Theorem 4.1 is complete.


## 5. Auxiliary Examples

Example 5.1.
Suppose thatf=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
(5.1)lim⁡z→0|λD(f(z))λD(z)fz|=lim⁡z→01-r21-r2/K(1/K+1)|z|1/K-12=∞,lim⁡z→0|f(z)-f(0)||z-0|=lim⁡z→0|z|1/K-1=∞,lim⁡z→0dh(f(0),f(z))dh(0,z)=lim⁡z→0log⁡⁡((1+|z|1/K)/(1-|z|1/K))log⁡⁡((1+|z|)/(1-|z|))=lim⁡z→0|z|1/K-1=∞,lim⁡z→0|λD*(f(z))λD*(z)fz|=lim⁡z→0⁡|z|log⁡⁡(1/|z|)|z|1/Klog⁡⁡(1/|z|1/k)(1/K+1)|z|1/k-12=K+12.Example 5.2.
Suppose thatf=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 that(5.2)lim⁡z→0|λD(f(z))λD(z)fz|=lim⁡z→0K+121-r21-r2KrK-1=0,lim⁡z→0⁡|z-0||f(z)-f(0)|=lim⁡z→0⁡|z|1-K=∞,lim⁡z→0dh(0,z)dh(f(0),f(z))=lim⁡z→0log⁡⁡((1+|z|)/(1-|z|))log⁡⁡((1+|z|K)/(1-|z|K))=lim⁡z→0⁡|z|1-K=∞,lim⁡z→0|λD*(f(z))λD*(z)fz|=lim⁡z→0|z|log⁡⁡(1/|z|)|z|Klog⁡(1/|z|K)K+12|z|K-1=K+12K.Example 5.3.
Suppose thatf(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,
(5.3)lim⁡|z|→∞⁡|fz|=lim⁡|z|→∞K+12K|z|K-1=+∞,|(log⁡⁡φ(w))w|=|φwφ|=|1w|⟶∞,w⟶0,lim⁡z→∞⁡|f(z)||z|=lim⁡z→∞|z|K-1=∞,‖∂f‖=K+12.Example 5.4.
LetΩ*=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
(5.4)|(log⁡φ(w)w)|=|φwφ|=|1w|≤1,lim⁡z→0‖∂f‖=lim⁡z→∞r2-1r2/K-1(1/K+1)|z|1/K-12=∞,lim⁡r→∞log⁡((1+1/r1/K)/(1-1/r1/K))log⁡((1+1/r)/(1-1/r))=∞.Example 5.5.
Let𝕌+ be the right half plane. Let Ω~=𝕌+∖[1,+∞). Then Ω~ is not an angular domain. The hyperbolic metric λΩ~(z)|dz| with gaussian curvature -4 is given by
(5.5)λΩ~(z)|dz|=1z2+1+z2+1¯|zz2+1||dz|.
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
(5.6)lim⁡z→0‖∂f‖=lim⁡z→0K+12K|z2+1||w2+1|z2+1+z2+1¯w2+1+w2+1¯|z|2(1/K-1)=∞,
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
(5.7)lim⁡z→0‖∂g‖=lim⁡z→0K+12|z2+1||ξ2+1|z2+1+z2+1¯ξ2+1+ξ2+1¯|z|2(K-1)=0,
where ξ=g(z).



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