-Harmonic Quasiconformal Mappings

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Volume 2012, Article ID 569481,13pages doi:10.1155/2012/569481

Research Article

Hyperbolically Bi-Lipschitz Continuity for 1/ | w |

²

-Harmonic Quasiconformal Mappings

Xingdi Chen

Department of Mathematics, Huaqiao University, Fujian, Quanzhou 362021, China

Correspondence should be addressed to Xingdi Chen,[email protected] Received 25 March 2012; Accepted 23 May 2012

Academic Editor: Oscar Blasco

Copyrightq2012 Xingdi Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the class of 1/|w|²-harmonicK-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 constantK.

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 homeomorphismfofΩontoΩis said to be aρ-harmonic mapping if it satisfies the Euler-Lagrange equation

f_zz logρ

w

f

f_zf_z0, 1.1

wherewfzandρw|dw|²is a smooth metric inΩ. Ifρis a constant thenfis said to be euclidean harmonic. A euclidean harmonic mapping defined on a simply connected domain is of the formfhg, wherehandgare two analytic functions inΩ. 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

fz

af f

fz, 1.2

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whereazis analytic and|az|<1see 4–6for more details. By diﬀerentiating1.2inz, we have that

fzz

log 1

|w|²

w

◦ffzfzaf f

fzz

log 1

|w|²

w

◦ffzfz

. 1.3

Hence, it follows that a logharmonic mapping is a 1/|w|²-harmonic mapping.

If aρ-harmonic mappingfalso satisfies the condition that|fzz| ≤k|fzz|holds for everyz∈Ω, then it is called aρ-harmonicK-quasiconformal mappingfor simplicity, a harmonic quasiconformal mapping or H.Q.C mapping, whereK 1k/1−k.

Let λ_Ωz|dz| denote the hyperbolic metric of a simply connected region Ω with gaussian curvature−4. For a harmonic quasiconformal mappingfofΩontoΩ, we call the quantity

∂f λ_Ω◦f

λ_Ω fz 1.4

the hyperbolically partial derivative off. Iffis a harmonic quasiconformal mapping ofΩ1onto Ω2andϕis a conformal mapping ofΩ0ontoΩ1thenf◦ϕis also a harmonic quasiconformal mapping. We have

∂

f◦ϕ λ_Ω₂

f◦ϕζ λ_Ω₀ζ

f◦ϕ

ζ λ_Ω₂ fz

λ_Ω₁z fz∂f, 1.5 wherezϕζ. Hence, we always fix the domain of a harmonic quasiconformal mapping to be the unit diskDwhen studying its hyperbolically partial derivative.

The hyperbolic distancedhz1, z2between z1 andz2 is defined by infγ

γλ_Ωz|dz|, where γ runs through all rectifiable curves in Ω which connect z₁ and z₂. A harmonic quasiconformal mappingfofΩontoΩis said to be hyperbolicallyL₁-LipschitzL1>0if

d_h

fz1, fz2

≤L₁d_hz1, z₂, z₁, z₂∈Ω. 1.6 The constantL1 is said to be the hyperbolically Lipschitz coeﬃcient off. If there also exists a constantL₂>0 such that

L₂d_hz1, z₂ ≤d_h

fz1, 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 off.

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. LetΩandΩbe two Jordan domains, letα∈0,1and letf :Ω→Ωbe a euclidean harmonic quasiconformal mapping. If∂Ωand∂Ω∈C^1,α, thenfis euclidean Lipschitz. In particular, ifΩis convex, thenfis euclidean bi-lipschitz.

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Recently, the hyperbolically Lipschitz or bi-lipschitz continuity of euclidean harmonic quasiconformal mappings also excited much interestsee 14–17. In 14, Chen and Fang proved the following.

Theorem B. Letfbe a euclidean harmonicK-quasiconformal mapping ofΩonto a convex domain Ω. Thenfis hyperbolically1/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 generalρ-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, and5.2show that if the metricρis not necessary to be smooth in the range of aρ-harmonic quasiconformal mappingf, thenfgenerally 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 Mateljevi´csee Theorem 4.4 of 18showed the following.

Theorem C. Letϕbe analytic inΩandfa|ϕ|-harmonic quasiconformal mapping of theC^1,αdomain Ωonto theC^1,αJordan domainΩ. IfMlogϕ_∞<∞, thenfis euclidean Lipschitz.

Let|ϕw|be equal to 1/|w|², wherew ∈ Ω. If the closure of the rangeΩdoes not include the origin, thenM logϕ_∞ 1/|w|_∞ is finite. So by Theorem C a 1/|w|²- harmonic quasiconformal mapping with such a rangeΩhas 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, Example5.3also 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 planeH, we first build a diﬀerential equation for the hyperbolic metric of an angular domain with the origin of Cas 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 Mateljevi´c 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 rangessee Theorem3.1. We also show that both the upper and lower bounds of∂fare 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 Theorem4.1. The hyperbolically bi-lipschitz coeﬃcients1/K, Kare sharp.

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At last, some auxiliary examples are given. In order to show the sharpness of Theorems 3.1and 4.1, we present two examples satisfying that the inequalities3.2 no longer hold for two classes of 1/|w|²-harmonic quasiconformal mappings with nonangular rangessee Examples5.4and5.5.

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

LetλHw|dw|be the hyperbolic metric of the upper half planeHwith gaussian curvature

−4. Then

λHw|dw| i

w−w|dw|, logλH

w− 1

w−w, logλH

ww 1

w−w². 2.1

Hence, the hyperbolic metricλHw|dw|ofHsatisfies that

logλH

ww

logλH

w

w w

wλ²_H 0. 2.2

By the relation thatlogλ_H_w λH_w/λ_H, the diﬀerential equation2.2becomes

λH_ww

λH λH_w λH

2

−λH_w wλH −w

wλH². 2.3

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

Lemma 2.1. LetAbe an angular domain with the origin of the complex planeCas its vertex. Then for everyζ∈Athe hyperbolic metricλAζ|dζ|ofAsatisfies the following diﬀerential equation

logλ_A

ζζ

logλA

ζ

ζ ζ

ζλA²0. 2.4

Proof. LetAθbe the angular domain{z∈C|0<arg z < θ, θ∈0,2π}with 0 as its vertex and λ_A_θz|dz|as its hyperbolic metric with gaussian curvature −4. Let f be a conformal mapping ofA_θontoH. Then by the fact that a hyperbolic metric is a conformal invariant it follows that

λ_A_θz λ_H◦ff. 2.5

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Hence by the chain rule 20we get logλ_A_θ

z λH_w◦ff λH◦f 1

2 f f, logλAθ

zz λH◦f

λH_ww◦ff² λH_w◦ff

−

λH_w◦ff₂ λ_H◦f2 f

2f

.

2.6

From the relations2.5and2.6we get logλ_A_θ

zz

logλ_A_θ

z

z z

zλAθ² λH_ww

λ_H ◦ff²λH_w

λ_H ◦ff−

λH_w λ_H ◦f

2

f² 1

2 f

f

λH_w λH ◦ff

z 1 2z

f f z

z

λ_H◦ff².

2.7

Using2.3we can simplify the previous relation as logλ_A_θ

zz

logλ_A_θ

z

z z

zλAθ² λH_w

λ_H ◦f

ff z −f²

f

1 2

f f

1 2z

f f

λ_H◦f₂ z

zf²− f ff²

, 2.8 wherewfz.

Letfz z^α, α ∈ 1/2,1∪1,∞. Then f is a conformal mapping ofAθonto the upper half planeHand the following relations

ff z − f²

f 0, 1 2

f f

1 2z

f

f 0, z

zf²− f

ff²0 2.9

hold for everyz∈Aθ. Hence, it follows from the above relations2.8and2.9that logλAθ

zz

logλAθ

z

z z

zλAθ²0. 2.10

LetAbe an arbitrary angular domain only satisfying that its vertex is the origin ofC.

Then there exists a rotation transformationzgζ e^iθ⁰ζ, ζ∈Awith 0≤θ₀ ≤2πsuch that gconformally mapsAontoA_θ. Hence,

λAζ λAθ

gζ ,

logλAζ

ζe^iθ⁰

logλθz

z,

logλAζ

ζζe^2iθ⁰

logλθz

zz. 2.11

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Thus by the relation2.10the following diﬀerential equation:

logλA

ζζ

logλA

ζ

ζ ζ

ζλA²0 2.12

holds for everyζ∈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 constantK, of its hyperbolically partial derivative.

To do so we need the well-known Ahlfors-Schwarz lemma 21and its opposite type given by Mateljevi´c 19as follows.

Lemma A. Ifρ >0 is aC² metric density onDfor which the gaussian curvature satisfiesK_ρ≥ −4 and ifρztends to∞when|z|tends to 1⁻, thenλD≤ρ.

Kalaj 7obtained the following.

Lemma B. LetΩbe a convex domain inC. Iffis a euclidean harmonicK-quasiconformal mapping of the unit disk ontoΩ, satisfyingf0 a, then

fz≥ 1

21kδ_Ω, z∈D, 3.1

whereδ_Ω da, ∂Ω inf{|f−a|:f∈∂Ω}andk K−1/K1.

Theorem 3.1. LetAbe an angular domain with the origin of the complex planeCas its vertex. Iff is a 1/|w|²-harmonicK-quasiconformal mapping of the unit diskDontoA, then for everyz∈Dits hyperbolically partial derivative satisfies the following inequality:

K1

2K ≤∂f≤ K1

2 . 3.2

Moreover, the upper and lower bound is sharp.

Proof. LetAbe an angular domain with the origin of the complex planeCas its vertex andf a 1/|w|²-harmonicK-quasiconformal mapping ofDontoA. Letk K−1/K1. From the assumptions we have thatfdoes not vanish onD. So logfis harmonic inΩ. Hence, we have thatlogf_zdoes not vanish by Lewy Theorem 22. Sof_zalso does not vanish. Suppose that σz 1−kλAfz|fz|,z∈D. Thereforeσz>0 for every pointz∈D. Thus we obtain

Δlogσ

z 4

logλ_A◦f

zzz

logf_z

zz

. 3.3

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By the chain rule 20we get

4

logλ_A◦f

zzz 4 logλ_A

ww◦ff_z²f_z² 2

logλ_A

ww◦f f_zf_z

2 logλ_A

w◦ff_zz .

3.4

By Euler-Lagrange equation we have that a 1/|w|²-harmonic mappingfsatisfies

f_zz−f_zf_z

f 0. 3.5

Sincefzdoes not vanish, we have from3.5that logf_z

zz 0. 3.6

Using the relations3.3,3.4,3.5, and3.6we have Δlogσ

z 4

logλ_A

ww◦ff_z²f_z² 2

logλ_A

ww

logλ_A

w

◦ff_zf_z

. 3.7

By the diﬀerential equation at Lemma2.1the above relation becomes

Δlogσ z 4

logλ_A

ww◦ff_z²f_z²

−2

λ_A◦f₂ffzfz

f

. 3.8

So we get

−Δlogσ

σ² −4 1−k²

Δlogλ_A

4λA² ◦ffz²fz²

fz² −2ff_z ff_z

. 3.9

By1.2it is clear that|fz/f_z||a|. Hence, it follows from3.9and the inequality|a| ≤kthat

K_σ −Δlogσ

σ² ≤ − 4 1−k²

1|a|²−2|a|

−41− |a|²

1−k² ≤ −4. 3.10

Thus by Ahlfors-Schwarz Lemma 21, P13it follows thatσ≤λD, that is, ∂f λ_A◦f

λ_D fz≤ K1

2 . 3.11

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LetF w|w|^K−1,w ∈H. ThenFis a 1/|w|²-harmonicK-quasiconformal mapping of Honto itself. Moreover, we also have

∂F λH◦F

λ_H |Fw| K1

2 . 3.12

ChoosingLto be a conformal mapping ofDontoH, we have thatF◦Lis 1/|w|²-harmonic K-quasiconformal mapping ofDontoH. Thus by1.5the equality3.12becomes that

∂F◦L K1

2 . 3.13

Therefore the upper bound at3.2is sharp.

Next we will prove the lower bound of∂f. Suppose that f is a 1/|w|²-harmonic K-quasiconformal mapping ofDontoA. Letδ 1kλAf|fz|.

Hence, we have

Δlogδ

z 4

logλA◦f

zzz

logfz

zz

. 3.14

Combining Lemma2.1with the relations3.4,3.5,3.6, and3.14we have

−Δlogδ

δ² −4 1k²

Δlogλ_A

4λA² ◦ffz²fz²

fz² −2ff_z ff_z

. 3.15

Hence, it follows from the inequality|a| ≤kand3.15that

K_δ −Δlogδ

δ² ≥ − 4 1k²

1|a|²2|a|

−41|a|²

1k² ≥ −4. 3.16

∂f λ_A◦f

λ_D f_z≥ K1

2K . 3.17

LetF w|w|^1/K−1,w∈H. ThenFis a 1/|w|²-harmonicK-quasiconformal mapping of Honto itself. Moreover, we also have

∂F λH◦F

λ_H |Fw| K1

2K . 3.18

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ChoosingLto be a conformal mapping ofDontoH, we have thatF◦Lis 1/|w|²-harmonic K-quasiconformal mapping ofDontoH. Thus by1.5it shows that

∂F◦L K1

2K . 3.19

Therefore the positive lower bound at3.2is 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. LetAbe an angular domain with the origin of the complex planeCas its vertex. Iff is a 1/|w|²-harmonicK-quasiconformal mapping of the unit diskDontoA, thenfis hyperbolically 1/K, K-bi-lipschitz. Moreover, both the coeﬃcientsKand 1/Kare sharp.

Proof. Letγbe the hyperbolic geodesic betweenz1andz2, wherez1andz2are two arbitrary points inD. Then it follows that

fγλAw|dw| ≤

γ

λA

fz

Lfz|dz| ≤ 2K K1

γ

λ_A

fzf_zz

λ_Dz λDz|dz|, 4.1

wherew fz. By the inequality of3.2and the definition of a hyperbolic geodesic, we obtain from the above inequality that

dh

fz1, fz2

≤

fγλAw|dw| ≤K

γ

λDz|dz|Kdhz1, z2. 4.2

Hence,fis hyperbolicallyK-Lipschitz.

LetF w|w|^K−1,w ∈H. ThenFis a 1/|w|²-harmonicK-quasiconformal mapping of Honto itself. Letz1iandz2iy,y >1 be two points inH. ThenFz1 iandFz2 iy^K. Thusd_hz1, z₂ logyandd_hFz1, Fz2 Klogy. So the equality

d_hFz1, Fz2 Kd_hz1, z₂ 4.3

holds. ChoosingLto be a conformal mapping ofDontoH, we have thatφF◦Lis 1/|w|²- harmonicK-quasiconformal mapping ofDontoH. Letφζ1 z1 andφζ2 z2. Thus by the fact that the hyperbolic distance is a conformal invariant it follows from1.5that

d_h

φζ1, φζ2

Kd_hLζ1, Lζ2 Kd_hζ1, ζ₂. 4.4 Thus the coeﬃcientKis sharp.

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Let fγ ⊂ A be the hyperbolic geodesic connected fz1 with fz2. By the assumption thatλ_A|fz|tends to∞as|z| → 1⁻, we have that the inequality3.2also holds.

Hence, we also have dh

fz1, fz2

fγλAw|dw| ≥ 1 K

γ

λDz|dz| ≥ 1

Kdhz1, z2, 4.5

wherewfz. Thusfis hyperbolically1/K, K-bi-lipschitz.

LetG w|w|^1/K−1,w ∈ H. Letz1 i, andz2 iy,y > 1 be two points inH. Then Gz1 iandGz2 iy^1/K. Thusd_hz1, z₂ logyandd_hGz1, Gz2 1/Klogy. So the equality

dhGz1, Gz2 dhz1, z2

K 4.6

holds. ChoosingLto be a conformal mapping ofDontoH, we have thatψ G◦Lis 1/|w|²- harmonicK-quasiconformal mapping ofDontoH. Letψζ1 z1andψζ2 z2. Thus by the fact that the hyperbolic distance is a conformal invariant it shows that

d_h

ψζ1, ψζ2

d_hLζ1, Lζ2

K d_hζ1, ζ₂

K . 4.7

Thus the coeﬃcient 1/Kis also sharp. The proof of Theorem4.1is complete.

5. Auxiliary Examples

Example 5.1. Suppose thatf z|z|^1/K−1, K > 1. LetD^∗ {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 onD. We have that f is a 1/|w|²-harmonic K-quasiconformal mapping ofD^∗ onto itself. If aρ-harmonic mapping is not necessary to be smooth, thenf is also a 1/|w|²- harmonicK-quasiconformal mapping ofDonto itself. Moreover, it follows that

z→lim0

λ_D fz λ_Dz fz

lim

z→0

1−r² 1−r^2/K

1/K1|z|^1/K−1

2 ∞,

z→lim0

fz−f0

|z−0| lim

z→0|z|^1/K−1∞,

z→lim0

d_h

f0, fz d_h0, z lim

z→0

log

1|z|^1/K /

1− |z|^1/K log1|z|/1− |z| lim

z→0|z|^1/K−1 ∞,

z→lim0

λ_D^∗ fz λ_D^∗z fz

lim

z→0

|z|log1/|z|

|z|^1/Klog

1/|z|^1/k1/K1|z|^1/k−1

2 K1

2 .

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 ofD^∗ onto itself. If aρ-harmonic mapping is not necessary to be

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smooth, thenfis also a 1/|w|²-harmonicK-quasiconformal mapping ofDonto itself. Similar to Example5.1, it follows that

z→lim0

λD

fz λ_Dz f_z

lim

z→0

K1 2

1−r²

1−r^2Kr^K−10,

z→lim0

|z−0|

fz−f0 lim

z→0|z|^1−K∞,

z→lim0

d_h0, z d_h

f0, fz lim

z→0

log1|z|/1− |z|

log

1|z|^K /

1− |z|^K lim

z→0|z|^1−K∞,

z→lim0

λ_D∗ fz λ_D^∗z fz

lim

z→0

|z|log1/|z|

|z|^Klog

1/|z|^KK1

2 |z|^K−1 K1 2K .

5.2

Example 5.3. Suppose thatfz z|z|^K−1, K >1. Thenfis a|ϕ|-harmonicK-quasiconformal mapping of the upper half planeHonto itself, hereϕw 1/w². Moreover,

|z| → ∞limf_z lim

|z| → ∞

K1

2K |z|^K−1 ∞, logϕw

w ϕ_w

ϕ

1 w

−→ ∞, w−→0,

zlim→ ∞

fz

|z| lim

z→ ∞|z|^K−1∞, ∂f K1 2 .

5.3

Example 5.4. Let Ω^∗C\D

{∞}andK >1. Letϕw 1/w²,w∈Ω^∗. Thenf z|z|^1/K−1 is a|ϕ|-harmonicK-quasiconformal mapping ofΩ^∗onto itself and satisfies that

logϕw_w ϕ_w

ϕ

1 w

≤1,

z→lim0∂f lim

z→ ∞

r²−1 r^2/K−1

1/K1|z|^1/K−1

2 ∞,

rlim→ ∞

log

11/r^1/K /

1−1/r^1/K log11/r/1−1/r ∞.

5.4

Example 5.5. LetUbe the right half plane. LetΩ U\ 1,∞. ThenΩ is not an angular domain. The hyperbolic metricλ_Ωz|dz|with gaussian curvature−4 is given by

λ_Ωz|dz| 1

√z²1√ z²1

z

√z²1

|dz|. 5.5

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Letfz z|z|^1/K−1,z ∈ Ω, where K > 1. Thenf is a 1/|w|²-harmonicK-quasiconformal mapping ofΩ onto itself. Moreover, we have

z→lim0∂f lim

z→0

K1 2K

√ z²1 √

w²1

√z²1√ z²1

√w²1√ w²1

|z|^21/K−1∞, 5.6

where w fz. Letgz z|z|^K−1, z ∈ Ω, where K > 1. Theng is a 1/|w|²-harmonic K-quasiconformal mapping ofΩ onto itself. Moreover, we have

z→lim0

∂g lim

z→0

K1 2

√ z²1

ξ²1

√z²1√ z²1 ξ²1

ξ²1

|z|^2K−10, 5.7

whereξgz.

Acknowledgments

Foundation items Supported by NNSF of China11101165, the Fundamental Research Funds for the Central Universities of Huaqiao universityJB-ZR1136and NSF of Fujian Province 2011J01011.

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-Harmonic Quasiconformal Mappings

Research Article

Hyperbolically Bi-Lipschitz Continuity for 1/ | w |

-Harmonic Quasiconformal Mappings

Xingdi Chen

1. Introduction

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

3. Sharp Bounds for Hyperbolically Partial Derivatives

4. Sharp Coefficients of Hyperbolically Lipschitz Continuity

5. Auxiliary Examples

Acknowledgments

References

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