Hyperbolically Bi Lipschitz Continuity for Harmonic Quasiconformal Mappings

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

fzz

logρ_wffzfz0, 1.1

wherewfzandρw|dw|2_{is 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

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 λΩz|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 λΩ◦f

λΩ

_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 λΩ2

f◦ϕζ

λΩ0ζ

f◦ϕ_ζ λΩ2

fz

λΩ1z

_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

γλΩz|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

dh

fz1, fz2≤L1dhz1, 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

L2dhz1, z2 ≤dh

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 quasicon-formal 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∈C1,α_{, thenfis euclidean Lipschitz. In particular,}

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 theC1,αdomain

Ωonto theC1,α_{Jordan 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

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

wλ

2

H 0. 2.2

By the relation thatlogλH_w λH_w/λH, the differential equation2.2becomes

λHww

λH

λHw

λH 2

−λHw

wλH −

w

wλH

2_{. 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

logλA

ζζ

logλA_ζ

ζ

ζ

ζλA

2_{0. 2.4}

Proof. LetAθbe the angular domain{z∈C|0<arg z < θ, θ∈0,2π}with 0 as its vertex

and λAθz|dz|as its hyperbolic metric with gaussian curvature −4. Let f be a conformal

mapping ofAθontoH. Then by the fact that a hyperbolic metric is a conformal invariant it

follows that

λAθz λH◦ff

Hence by the chain rule 20we get

logλAθ

z

λH_w◦ff

λH◦f

1 2

f

f,

logλAθ

zz

λH◦fλH_ww◦ff2 λH_w◦ff−λH_w◦ff2

λH◦f

2 f

2f

.

2.6

From the relations2.5and2.6we get

logλAθ

zz

logλAθ

z

z

z

zλAθ

2

λHww

λH ◦

ff2λHw

λH ◦

ff−

_λH

w

λH ◦

f 2

f2

1 2 f f λHw

λH ◦f

f z 1 2z f f z z

λH◦ff2.

2.7

Using2.3we can simplify the previous relation as

logλAθ

zz

logλAθ

z

z

z

zλAθ

2

λHw

λH ◦

f

ff

z −

f2

f 1 2 f f 1 2z f f

λH◦f

2

z

zf

2₋ f

ff

2

,

2.8

wherewfz.

Letfz zα_{, 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

zf

2₋ f

ff

2

0 2.9

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

logλAθ

zz

logλAθ

z

z

z

zλAθ

2_{0. 2.10}

LetAbe an arbitrary angular domain only satisfying that its vertex is the origin ofC. Then there exists a rotation transformationzgζ eiθ0_{ζ, ζ}∈_{Awith 0}≤_θ

0 ≤2πsuch that gconformally mapsAontoAθ. Hence,

λAζ λAθ

gζ, logλAζ_ζeiθ0_logλθz

z,

logλAζ_ζζe2iθ0_logλθz

zz.

Thus by the relation2.10the following differential equation:

logλA_ζζ

logλA_ζ

ζ

ζ

ζλA

2_{0 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

21kδΩ, z∈D, 3.1

whereδΩ 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. Sofzalso does not vanish. Suppose that σz 1−kλAfz|fz|,z∈D. Thereforeσz>0 for every pointz∈D. Thus we obtain

Δlogσz 4logλA◦f

zzz

logfz_zz

By the chain rule 20we get

4logλA◦f

zzz 4

logλA

ww◦ffz

2

fz2

2logλA

ww◦f

fzfz

2logλA

w◦ffzz

.

3.4

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

fzz−

fzfz

f 0. 3.5

Sincefzdoes not vanish, we have from3.5that

logfz_zz 0. 3.6

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

Δlogσz 4

logλA

ww◦f

fz2fz2

2

logλA

ww

logλA

w w

◦ffzfz

.

3.7

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

Δlogσz 4

logλA

ww◦f

fz2fz2

−2

λA◦f

2ffzfz

f

. 3.8

So we get

−Δlogσ

σ2

−4

1−k2

ΔlogλA

4λA2 ◦f

_fz2

fz2

_fz2 −2

ffz

ffz

. 3.9

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

Kσ −

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 λA◦f

λD

fz≤ K1

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

λH |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

Δlogδz 4logλA◦f_zzz logfz_zz. 3.14

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

−Δlogδ

δ2

−4

1k2

ΔlogλA

4λA2 ◦f

_fz2

fz2

_fz2 −2

ffz

ffz

. 3.15

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

Kδ −

Δlogδ

δ2 ≥ −

4

1k2

1|a|22|a|−41|a|

2

1k2 ≥ −4. 3.16

Since the mapping logw mapsAonto a strip domainS, we have that logf is an euclidean harmonic mapping ofDontoS. So it follows from Lemma B that|logf_z| ≥C0, whereC0is a positive constant. Thus we haveλAf|fz|λSlogf|logf_z| → ∞as|z| → 1−. Thus it follows from Lemma A that

∂f λA◦f

λD

fz≥

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

λH |Fw|

K1

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

λAfzLfz|dz| ≤ 2K

K1

γ

λA

fzfzz

λDz λDz|dz|, 4.1

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

dhfz1, 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 iyK_.

Thusdhz1, z2 logyanddhFz1, Fz2 Klogy. So the equality

dhFz1, Fz2 Kdhz1, 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

dh

φζ1, φζ2KdhLζ1, Lζ2 Kdhζ1, ζ2. 4.4

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

dhfz1, 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 iy1/K_{. Thusdhz}

1, z2 logyanddhGz1, 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

dh

ψζ1, ψζ2 dhLζ1, Lζ2

K

dhζ1, 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

lim

z→0

λD

fz

λDz fz

z→lim0

1−r2

1−r2/K

1/K1|z|1/K−1

2 ∞,

lim

z→0

fz−f0

|z−0| z→lim0|z|

1/K−1_∞,

lim

z→0

dh

f0, fz

dh0, z z→lim0

log1|z|1/K/1− |z|1/K

log1|z|/1− |z| z→lim0|z|

1/K−1 _∞,

lim

z→0

λD∗

fz

λD∗z

fz lim

z→0

|z|log1/|z|

|z|1/K_log1/|z|1/k

1/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}

smooth, thenfis also a 1/|w|2_{-harmonicK-quasiconformal mapping ofDonto itself. Similar}

to Example5.1, it follows that

lim

z→0

λDfz

λDz fz

z→lim0

K1

2

1−r2

1−r2Kr

K−1_0,

lim

z→0

|z−0|

_fz−f0 zlim→0|z|

1−K_∞,

lim

z→0

dh0, z

dh

f0, fz zlim→0

log1|z|/1− |z|

log1|z|K/1− |z|K zlim→0|z|

1−K_∞,

lim

z→0

λD∗fz

λD∗z

fzlim

z→0

|z|log1/|z|

|z|K_log1/|z|K

K1

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,}

lim

|z| → ∞fz|z| → ∞lim

K1

2K |z|

K−1 _{∞, logϕw}

w

ϕw

ϕ

1

w

−→ ∞, w−→0,

lim

z→ ∞

_fz

|z| z→ ∞lim|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_wϕw

ϕ

1

w

≤1,

lim

z→0∂fzlim→ ∞

r2₋₁

r2/K−1

1/K1|z|1/K−1

2 ∞,

lim

r→ ∞

log11/r1/K_/1−1/r1/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

Letfz z|z|1/K−1,z ∈ Ω, whereK > 1. Thenf is a 1/|w|2_{-harmonicK-quasiconformal}

mapping ofΩ onto itself. Moreover, we have

lim

z→0∂fz→lim0

K1

2K

√z2₁

√w2₁

√

z2₁√_z2₁

√

w2₁√_w2₁

|z|21/K−1_{∞, 5.6}

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

lim

z→0

∂g lim

z→0

K1

2

√z2₁

ξ2₁

√

z2₁√_z2₁

ξ2_1ξ2₁

|z|2K−1_{0, 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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