Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk
UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem,  and coefficient estimate for the classes of $(K,K')$-quasiconformal harmonic mappings from the unit disk onto itself.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512244822966272 |
|---|---|
| author | Zhong, Deguang Yuan, Wenjun Deguang Yuan, Wenjun Zhong, Deguang Yuan, Wenjun |
| author_facet | Zhong, Deguang Yuan, Wenjun Deguang Yuan, Wenjun Zhong, Deguang Yuan, Wenjun |
| author_sort | Zhong, Deguang |
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| datestamp_date | 2025-03-31T08:48:28Z |
| description | UDC 517.51
We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem,  and coefficient estimate for the classes of $(K,K')$-quasiconformal harmonic mappings from the unit disk onto itself.
|
| doi_str_mv | 10.37863/umzh.v73i2.6041 |
| first_indexed | 2026-03-24T03:25:43Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i2.6041
UDC 517.51
Deguang Zhong (Dep. Appl. Statistics, Guangdong Univ. Finance, Guangzhou, China),
Wenjun Yuan (Dep. Basic Course Teaching, Software Engineering Inst. Guangzhou, China)
HYPERBOLICALLY LIPSCHITZ CONTINUITY, AREA DISTORTION
AND COEFFICIENT ESTIMATES FOR (\bfitK ,\bfitK \prime )-QUASICONFORMAL
HARMONIC MAPPINGS OF UNIT DISK
ГIПЕРБОЛIЧНА НЕПЕРЕРВНIСТЬ ЗА ЛIПШИЦЕМ, СПОТВОРЕННЯ
ОБЛАСТЕЙ ТА ОЦIНКИ КОЕФIЦIЄНТIВ ДЛЯ (\bfitK ,\bfitK \prime )-КВАЗIКОНФОРМНИХ
ГАРМОНIЧНИХ ВIДОБРАЖЕНЬ ОДИНИЧНОГО ДИСКА
We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem, and coefficient estimate
for the classes of (K,K\prime )-quasiconformal harmonic mappings from the unit disk onto itself.
Вивчаються гiперболiчна неперервнiсть за Лiпшицем, теорема про спотворення евклiдових та гiперболiчних облас-
тей, а також оцiнки коефiцiєнтiв для (K,K\prime )-квазiконформних гармонiчних вiдображень одиничного диска в себе.
1. Introduction. Before stating some backgrounds and our main results, we firstly introduce some
terminologies. Suppose that \gamma is a rectifiable curve in the complex plane. Denote by l the length of
\gamma and let \Gamma : [0, 1] \mapsto \rightarrow \gamma be the natural parameterization of \gamma , i.e., the parameterization satisfying the
condition
| \.\Gamma (s)| = 1 for all s \in [0, 1].
We will say that \gamma is of class Cn,\mu for n \in \BbbN , 0 < \mu < 1, if \Gamma is of class Cn and
\mathrm{s}\mathrm{u}\mathrm{p}
t,s\in [0,1]
| \Gamma (n)(t) - \Gamma (n)(s)|
| t - s| \mu
< \infty .
We will call a Jordan Cn,\mu domain in \BbbC , if is bounded by Cn,\mu Jordan curve.
Let D and G be subdomains of the complex plane \BbbC . We say that a function u : D \mapsto \rightarrow \BbbR is
absolutely continuous on line in the region D if for every closed rectangle R \subset D with sides parallel
to the axes x and y, u are absolutely continuous on almost every horizontal line and almost every
vertical line in R. Such a function has, of course, partial derivatives ux and uy everywhere in D. A
topological mapping f = u+ iv : D \rightarrow G is said to be (K,K \prime )-quasiconformal if it satisfies:
(a) f is absolutely continuous on lines in D;
(b) there are constants K \geq 1 and K \prime \geq 0 such that
c\bigcirc DEGUANG ZHONG, WENJUN YUAN, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2 151
152 DEGUANG ZHONG, WENJUN YUAN
L2
f \leq KLf lf +K \prime , a.e. in D,
where Lf = | fz(z)| + | fz(z)| , lf = | | fz(z)| - | fz(z)| | , fz =
1
2
(fx - ify) and fz =
1
2
(fx + ify). If
K \prime = 0, then f is called a K -quasiconformal mapping.
Let \rho (z)| dz| 2 be a conformal C1 metric defined on D. A map f \in C2(D,G) is called a
\rho -harmonic mapping if
fzz + (\mathrm{l}\mathrm{o}\mathrm{g} \rho )\omega \circ f \cdot fzfz = 0.
In particular, 1-harmonic mapping is called an Euclidean harmonic function. In what follows, we say
a function is harmonic always means that it is Euclidean harmonic.
Let \lambda D(z)| dz| be the hyperbolic metric of the domain G having constant Gaussian curvature - 1.
The hyperbolic distance dhD
(z1, z2) between two points z1 and z2 in D is defined by
\mathrm{i}\mathrm{n}\mathrm{f}
\gamma
\left\{
\int
\gamma
\lambda D(z)| dz|
\right\} ,
where infimum is taken over all rectifiable curves \gamma in D connecting z1 and z2. It is known that if
D = \BbbD , then
\lambda \BbbD (z) =
2
1 - | z| 2
and dh\BbbD (z1, z2) = \mathrm{l}\mathrm{o}\mathrm{g}
| 1 - z1z2| + | z1 - z2|
| 1 - z1z2| - | z1 - z2|
.
A mapping h of D onto G is said to be hyperbolically Lipschitz if there exists a constant L1 > 0,
such that the inequality
dhG
(h(z1), h(z2)) \leq L1 dhD
(z1, z2)
holds for every z1, z2 \in D.
We will say that a mapping f : \BbbD \rightarrow \Omega is normalized if f(ti) = \omega i, i = 0, 1, 2, where
\{ t0t1, t1t2, t2t0\} and \{ \omega 0\omega 1, \omega 1\omega 2, \omega 2\omega 1\} are arcs of \BbbT = \partial \BbbD and of \gamma = \partial \Omega , respectively, having
the same length 2\pi /3 and | \gamma | /3, respectively. Let \gamma \in C1,\mu , 0 < \mu < 1, be a Jordan curve, g be the
arc length parameterization of \gamma and l = | \gamma | be the length of \gamma . Let d\gamma be the distance between g(s)
and g(t) along the curve \gamma , i.e.,
d\gamma (g(s), g(t)) = \mathrm{m}\mathrm{i}\mathrm{n}\{ | s - t| , (l - | s - t| )\} .
A closed rectifiable Jordan curve \gamma enjoys a b-chord-arc condition for some constant b > 1 if for all
z1, z2 \in \gamma there holds the inequality
d\gamma (z1, z2) \leq b| z1 - z2| . (1.1)
It is clear that if \gamma \in C1,\mu then \gamma enjoys a chord-arc condition for some b\gamma > 1.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
HYPERBOLICALLY LIPSCHITZ CONTINUITY, AREA DISTORTION AND COEFFICIENT ESTIMATES . . . 153
1.1. Background and main results. Martio [12] was the first one to consider quasiconformal
harmonic mappings for the unit disk, and Kalaj [4] extended the domain to the unit ball. In [14],
Wan showed that every hyperbolically harmonic quasiconformal diffeomorphism from \BbbD onto itself
is a quasiisometry of the Poincaré disk. In [11], Parlović proved that a K -quasiconformal harmonic
mapping of \BbbD onto itself is bi-Lipschitz with respect to Euclidean distence. Its explicit bi-Lipschitz
constants were given by Partyka and Sakan [5]. In 2007, Knežević and Matecljević [10] showed
that a K -quasiconformal harmonic mapping of the unit disk onto itself is a
\Bigl( 1
K
,K
\Bigr)
-quasiisometry
with respect to Poincaré metric. Recently, Kalaj and Mateljević [2] studied the class of (K,K \prime )-
quasiconformal mappings with bounded image domains. They obtained the following intrigue
results [2].
Theorem 1.1 [2]. Suppose that \Omega is a Jordan domain with C2 boundary and \omega is (K,K \prime )-
quasiconformal harmonic mapping between the unit disk \BbbD and \Omega . Then:
(a) \omega has a continuous extension to \BbbD , whose restriction to \BbbT we denote by f ;
(b) furthermore, \omega is Lipschitz continuous on \BbbD ;
(c) if f is normalized, there exists a constant L = L(K,K \prime , \partial \Omega ) such that
| f \prime (t)| \leq L for almost every t \in [0, 2\pi ],
and
| \omega (z1) - \omega (z2)| \leq (KL+
\surd
K \prime ) | z1 - z2| for z1, z2 \in \BbbD .
Here,
L \leq
\Bigl(
K\lambda k0b(L\lambda (K,K \prime ))1+1/\lambda \pi 1/\lambda +
\surd
K \prime
\Bigr) \lambda
, \alpha =
1
K(1 + 2b)2
, \lambda =
2 - \alpha
\alpha
, k0 = \mathrm{s}\mathrm{u}\mathrm{p}
s
| ks| ,
and ks is the curvature of \partial \Omega at the point g(s), b is a constant such that \partial \Omega satisfies b-chord-arc
condition in (1.1),
L\lambda (K,K \prime ) = 4(1 + 2b)2\alpha
\sqrt{}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2\pi K| \Omega |
\mathrm{l}\mathrm{o}\mathrm{g} 2
,
2\pi K \prime
K(1 + 2b)2 + 4
\biggr\}
.
The hyperbolically Lipschitz continuity for (K,K \prime )-quasiconformal harmonic mapping from
upper half-plane onto itself was obtained by Min Chen and Xingdi Chen (see [8], Theorem 2.2). The
first aim of this paper, we study the hyperbolically Lipschitz continuity for the class of (K,K \prime )-
quasiconformal harmonic mappings from unit disk onto itself. Our result reads as follows.
Theorem 1.2. Suppose that \omega is (K,K \prime )-quasiconformal harmonic mapping from unit disk onto
itself satisfying | \omega - 1(0)| \leq l < 1, where l is a constant, then \omega is hyperbolically Lipschitz continuity.
In 1994, Astala [8] proved that if f is a K -quasiconformal mapping from the unit disk \BbbD onto
itself, normalized by f(0) = 0, and if E is any measurable subset of the unit disk, then Ae(f(E)) \leq
\leq a(K)Ae(E)1/K , where Ae(\cdot ) denotes the Euclidean area and a(K) \rightarrow 1 when K \rightarrow 1+. In 1998,
Porter and Reséndis [13] obtained some results about area distortion under quasiconformal mappings
on the unit disk \BbbD onto itself with respect to the hyperbolic measure. They also showed the existence
of explodable sets; this kind of sets has bounded hyperbolic area, but under a specific quasiconformal
mapping its image has infinite hyperbolic area. In [1], Hernándezmontes and Reséndis studied the
hyperbolic and Euclidean area distortion of measurable sets under some classes of K -quasiconformal
mappings from the upper half-plane and the unit disk onto themselves, respectively. The Euclidean
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
154 DEGUANG ZHONG, WENJUN YUAN
and hyperbolic area distortion theorems for (K,K \prime )-quasiconformal harmonic mapping from upper
half-plane onto itself were obtained by Min Chen and Xingdi Chen [9]. It was showed by Kalaj and
Mateljević [2] (Example 1.5) that a (K,K \prime )-quasiconformal harmonic mapping from unit disk onto
itself is generally not a (K, 0)-quasiconformal harmonic mapping. So, it is interesting to study the
Euclidean and hyperbolic area distortion for (K,K \prime )-quasiconformal harmonic mapping from unit
disk onto itself. We have the following theorem.
Theorem 1.3. Let \omega be a (K,K \prime )-quasiconformal harmonic mapping from unit disk \BbbD onto
itself satisfying \omega (0) = 0. If \omega | \BbbT = f is normalized, then, for any measurable set E \subset \BbbD , we have
Ae(\omega (E)) \leq (K\phi (K,K \prime ) +
\surd
K \prime )2Ae(E)
and
A\scrH (\omega (E)) \leq \pi 2
4
(K\phi (K,K \prime ) +
\surd
K \prime )2A\scrH (E),
where Ae(\cdot ) and A\scrH (\cdot ) denote the Euclidean and hyperbolic area, respectively. Here,
\phi (K,K \prime ) =
\biggl(
\pi K[2K(1 + \pi )2 - 1]
2
\varphi (K,K \prime )\pi
1
2K(1+\pi )2 - 1 +
\surd
K \prime
\biggr) 2K(1+\pi )2 - 1
and
\varphi (K,K \prime ) =
\Biggl(
4(1 + \pi ) \cdot 21/K(1+\pi )2
\sqrt{}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2K\pi 2
\mathrm{l}\mathrm{o}\mathrm{g} 2
,
2\pi K \prime
K(1 + \pi )2 + 4
\biggr\} \Biggr) 2K(1+\pi )2
2K(1+\pi )2 - 1
.
In [7], Zhu obtained the coefficient estimates for K -quasiconformal harmonic mappings from
unit disk onto itself. Here we consider the coefficient estimates for the (K,K \prime )-quasiconformal
harmonic mappings of unit disk \BbbD onto itself. We have the following theorem.
Theorem 1.4. Given K \geq 1,K \prime \geq 0. Let \omega (z) = h(z) + g(z) be a (K,K \prime )-quasiconformal
harmonic mapping from unit disk onto itself satisfying \omega (0) = 0, where
h(z) =
\infty \sum
n=1
anz
n and g(z) =
\infty \sum
n=1
bnz
n
are analytic in \BbbD . If the boundary function f of \omega is normalized, then
| an| + | bn| \leq
4\phi (K,K \prime )
n\pi
, n = 1, 2, . . . ,
where
\phi (K,K \prime ) =
\biggl(
\pi K[2K(1 + \pi )2 - 1]
2
\varphi (K,K \prime )\pi
1
2K(1+\pi )2 - 1 +
\surd
K \prime
\biggr) 2K(1+\pi )2 - 1
and
\varphi (K,K \prime ) =
\Biggl(
4(1 + \pi ) \cdot 21/K(1+\pi )2
\sqrt{}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2K\pi 2
\mathrm{l}\mathrm{o}\mathrm{g} 2
,
2\pi K \prime
K(1 + \pi )2 + 4
\biggr\} \Biggr) 2K(1+\pi )2
2K(1+\pi )2 - 1
.
The remainder of this paper are devoted to prove Theorems 1.2, 1.3 and 1.4, which will be
presented in Sections 2, 3 and 4, respectively.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
HYPERBOLICALLY LIPSCHITZ CONTINUITY, AREA DISTORTION AND COEFFICIENT ESTIMATES . . . 155
2. Hyperbolically Lipschitz continuity. The aim of this section is to prove Theorem 1.2. We
need the following lemma which will be used in the proof of Theorem 1.2. (See [3], Remark 2.4, for
the case of a = 0.)
Lemma 2.1. Let \omega be a harmonic mapping from unit disk into itself satisfying \omega (a) = 0, then
1 - | z| 2
1 - | \omega (z)| 2
\leq \pi
2
1 + | a|
1 - | a|
. (2.1)
Proof. Let \varphi (z) =
z + a
1 + az
and F (z) = \omega (\varphi (z)), then F (z) is a harmonic mapping from unit
disk onto itself satisfying F (0) = 0. Hence by harmonic Schwarz lemma [5] (Lemma) and the
elementary inequality \bigm| \bigm| \bigm| \bigm| z - a
1 - az
\bigm| \bigm| \bigm| \bigm| \leq | z| + | a|
1 + | a| | z|
, z, a \in \BbbD ,
we have
| \omega (z)| \leq 4
\pi
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}
\bigm| \bigm| \bigm| \bigm| z - a
1 - az
\bigm| \bigm| \bigm| \bigm| \leq 4
\pi
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}
| z| + | a|
1 + | a| | z|
. (2.2)
Consider the function \varphi : [0, 1) - \rightarrow \BbbR and \varphi (t) =
4
\pi
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n} t - 2
\pi
(t - 1) - 1. As \varphi \prime (t) =
2
\pi
1 - t2
1 + t2
>
> 0, we get
4
\pi
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n} t \leq 2
\pi
(t - 1) + 1, t \in [0, 1). (2.3)
Combining (2.2) and (2.3), we have
1 - | \omega (z)| 2
1 - | z| 2
\geq
1 -
\biggl(
4
\pi
\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}
\biggl(
| z| + | a|
1 + | a| | z|
\biggr) \biggr) 2
1 - | z| 2
\geq
1 -
\biggl(
2
\pi
\biggl(
| z| + | a|
1 + | a| | z|
- 1
\biggr)
+ 1
\biggr) 2
1 - | z| 2
=
=
4
\pi
(1 - | a| )
\biggl[
1
(1 + | a| | z| )(1 + | z| )
- 1
\pi
(1 - | a| )(1 - | z| )
(1 + | a| | z| )2(1 + | z| )
\biggr]
. (2.4)
Let
\phi (t) =
1
(1 + tm)(1 + t)
- 1
\pi
(1 - m)(1 - t)
(1 + tm)2(1 + t)
, t,m \in [0, 1),
then
\phi \prime (t) = - 2tm+m+ 1
(1 + tm)2(1 + t)2
- 2(1 - m)
\pi
t2 - tm - m - 1
(1 + tm)3(1 + t)2
=
= -
(2tm+m+ 1)(1 + tm) +
2(1 - m)(t2m - tm - m - 1)
\pi
(1 + tm)3(1 + t)2
<
< - (2tm+m+ 1)(1 + tm) + (1 - m)(t2m - tm - m - 1)
(1 + tm)3(1 + t)2
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
156 DEGUANG ZHONG, WENJUN YUAN
= - t2m2 + 2tm2 + t2m+ tm+m2 +m
(1 + tm)3(1 + t)2
\leq 0.
Therefore, \phi (t) is monotonically decreasing on [0, 1), so we get
\phi (t) =
1
(1 + tm)(1 + t)
- 1
\pi
(1 - m)(1 - t)
(1 + tm)2(1 + t)
\geq \phi (1) =
1
2(1 +m)
. (2.5)
Thus, (2.1) is immediately derived from (2.4) and inequality (2.5).
Lemma 2.1 is proved.
Proof of Theorem 1.2. In order to prove Theorem 1.2, we only need to prove that
| \nabla \omega (z)| (1 - | z| 2)
1 - | \omega (z)| 2
< +\infty holds for every z \in \BbbD . Since \omega is harmonic from unit disk onto it-
self satisfying | \omega - 1(0)| \leq l < 1, hence, by Lemma 2.1, we have
1 - | z| 2
1 - | \omega (z)| 2
\leq \pi
2
1 + | \omega - 1(0)|
1 - | \omega - 1(0)|
\leq \pi
2
1 + l
1 - l
. (2.6)
Moreover, by Theorem 1.1, there exists a positive constant M such that the inequality
| \nabla \omega | \leq M (2.7)
holds, for every z \in \BbbD . Combining (2.6) and (2.7), we get
| \nabla \omega (z)| (1 - | z| 2)
1 - | \omega (z)| 2
\leq \pi M
2
1 + l
1 - l
< +\infty .
Theorem 1.2 is proved.
3. Area distortion. In this section, we will prove Theorem 1.3. In order to derive an explicit
Lipschitz constant in Theorem 1.2 in the setting of (K,K \prime )-quasiconformal harmonic mapping from
unit disk onto itself, we need the following lemma.
Lemma 3.1. \gamma = \partial \BbbD satisfies the
\pi
2
-chord-arc condition. Namely, for all g(s), g(t) \in \partial \BbbD ,
there exists b =
\pi
2
> 1 such that
d\gamma (g(s), g(t)) \leq
\pi
2
| g(s) - g(t)| . (3.1)
Proof. Suppose that g(s) = ei\alpha , g(t) = ei\beta \in \gamma = \partial \BbbD . Without loss of generality, we can
assume the angle, denoted by \theta , between g(s) and g(t) satisfies 0 < \theta < \pi . Namely, 0 < \theta =
= | \alpha - \beta | < \pi , then d\gamma (g(s), g(t)) = \theta . Since | g(s) - g(t)| = 2 \mathrm{s}\mathrm{i}\mathrm{n}
\theta
2
, by Jordan inequality, we
get
2
\pi
\theta
2
\leq \mathrm{s}\mathrm{i}\mathrm{n}
\theta
2
=
| g(s) - g(t)|
2
,
which yields (3.1).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
HYPERBOLICALLY LIPSCHITZ CONTINUITY, AREA DISTORTION AND COEFFICIENT ESTIMATES . . . 157
Proof of Theorem 1.3. Considering the case for \BbbD = \Omega in Theorem 1.1. By Lemma 3.1, we
have \alpha =
1
K(1 + \pi )2
and \lambda = 2K(1 + \pi )2 - 1. By Theorem 1.1, together with k0 = 1, we obtain
L \leq \phi (K,K \prime ), (3.2)
where
\phi (K,K \prime ) =
\biggl(
\pi K[2K(1 + \pi )2 - 1]
2
\varphi (K,K \prime )\pi
1
2K(1+\pi )2 - 1 +
\surd
K \prime
\biggr) 2K(1+\pi )2 - 1
and
\varphi (K,K \prime ) =
\Biggl(
4(1 + \pi ) \cdot 21/K(1+\pi )2
\sqrt{}
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
2K\pi 2
\mathrm{l}\mathrm{o}\mathrm{g} 2
,
2\pi K \prime
K(1 + \pi )2 + 4
\biggr\} \Biggr) 2K(1+\pi )2
2K(1+\pi )2 - 1
.
Using Theorem 1.1 again, we get
| \omega z(z)| \leq | \nabla \omega | \leq K \phi (K,K \prime ) +
\surd
K \prime , (3.3)
Combining (2.1) and (3.3), we have
| \omega z(z)|
1 - | \omega (z)| 2
\leq \pi
2
K \phi (K,K \prime ) +
\surd
K \prime
1 - | z| 2
. (3.4)
Furthermore, the Jacobian J\omega of \omega satisfies
J\omega = | \omega z(z)| 2 - | \omega z(z)| 2 \leq | \omega z(z)| 2 \leq
\Bigl(
K \phi (K,K \prime ) +
\surd
K \prime
\Bigr) 2
,
hence, for any measurable set E \subset \BbbD , we obtain
Ae(\omega (E)) =
\int
\omega (E)
dudv =
\int
E
J\omega (z)dxdy \leq
\Bigl(
K \phi (K,K \prime ) +
\surd
K \prime
\Bigr) 2
Ae(E).
In addition, by (3.4), we have
A\scrH (\omega (E)) =
\int
\omega (E)
4dudv
(1 - | \omega (z)| 2)2
=
\int
E
4J\omega (z)
(1 - | \omega (z)| 2)2
dxdy \leq
\int
E
4| \omega z(z)| 2
(1 - | \omega (z)| 2)2
dxdy \leq
\leq
\int
E
4
\Bigl[ \pi
2
(K \phi (K,K \prime ) +
\surd
K \prime )
\Bigr] 2
(1 - | z| 2)2
dxdy =
\pi 2
4
\Bigl(
K \phi (K,K \prime ) +
\surd
K \prime
\Bigr) 2
A\scrH (E).
Theorem 1.3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
158 DEGUANG ZHONG, WENJUN YUAN
4. Coefficient estimates. In this section, we will prove Theorem 1.4. We follow the idea in [7]
(Theorem 3).
Proof of Theorem 1.4. For every z = rei\theta \in \BbbD , \omega (rei\theta ) =
\sum \infty
n=1
anr
nein\theta +
\sum \infty
n=1
bnr
nein\theta .
Hence,
anr
n =
1
2\pi
2\pi \int
0
\omega (rei\theta )e - in\theta d\theta , n = 1, 2, . . . ,
and
bnr
n =
1
2\pi
2\pi \int
0
\omega (rei\theta )ein\theta d\theta , n = 1, 2, . . . .
For every n, setting an = | an| ei\alpha n , bn = | bn| ei\beta n , and \theta n =
\alpha n + \beta n
2n
. Then
(| an| + | bn| )rn =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 12\pi
2\pi \int
0
\omega (rei\theta )[e - i\alpha ne - in\theta + ei\beta nein\theta ]d\theta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 12\pi
2\pi \int
0
\omega (rei\theta )[e - in(\theta +\theta n) + ein(\theta +\theta n)]d\theta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1\pi
2\pi \int
0
\omega (rei\theta ) \mathrm{c}\mathrm{o}\mathrm{s}n(\theta + \theta n)d\theta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| .
Integrating by parts, we have
(| an| + | bn| )rn =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1n\pi
2\pi \int
0
\omega \theta (re
i\theta ) \mathrm{s}\mathrm{i}\mathrm{n}n(\theta + \theta n)d\theta
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| . (4.1)
By Theorem 1.1, we can see that f is absolutely continuous, hence,
\partial \omega
\partial \theta
(z) = P [f \prime ](z),
where P (r, x) =
1 - r2
2\pi (1 - 2r \mathrm{c}\mathrm{o}\mathrm{s}x+ r2)
and P [f ](z) =
\int 2\pi
0
P (r, x - \varphi )f(eix)dx. In addition, by
[2] (Lemma 4.1), the radial limits of \omega \theta exist almost everywhere and \mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow 1 - \omega \theta (re
i\theta ) = f \prime (\theta ).
Hence, tending r \rightarrow 1 - in (4.1) and also by Theorem 1.1 and (3.2), we obtain
| an| + | bn| \leq
1
n\pi
2\pi \int
0
| f \prime (\theta )| | \mathrm{s}\mathrm{i}\mathrm{n}n(\theta + \theta n)| d\theta \leq 4\phi (K,K \prime )
n\pi
.
Theorem 1.4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
HYPERBOLICALLY LIPSCHITZ CONTINUITY, AREA DISTORTION AND COEFFICIENT ESTIMATES . . . 159
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Received 08.05.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 2
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| id | umjimathkievua-article-6041 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:43Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/c0bbf3c37a60ae7b50d05b82ba1e4694.pdf |
| spelling | umjimathkievua-article-60412025-03-31T08:48:28Z Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk Zhong, Deguang Yuan, Wenjun Deguang Yuan, Wenjun Zhong, Deguang Yuan, Wenjun (K, K′ )-quasiconformal mapping hyperbolically Lipschitz continuous area distortion coefficient estimate (K, K′ )-quasiconformal mapping hyperbolically Lipschitz continuous area distortion coefficient estimate UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem,&nbsp; and coefficient estimate for the classes of $(K,K')$-quasiconformal harmonic mappings from the unit disk onto itself. УДК 517.51 Вивчаються гіперболічна неперервність за Ліпшицем, теорема про спотворення евклідових та гіперболічних облас\-тей, а також оцінки коефіцієнтів для $(K,K')$-квазіконформних гармонічних відображень одиничного диска в себе. Institute of Mathematics, NAS of Ukraine 2021-02-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6041 10.37863/umzh.v73i2.6041 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 2 (2021); 151 - 159 Український математичний журнал; Том 73 № 2 (2021); 151 - 159 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6041/8949 |
| spellingShingle | Zhong, Deguang Yuan, Wenjun Deguang Yuan, Wenjun Zhong, Deguang Yuan, Wenjun Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title | Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title_alt | Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title_full | Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title_fullStr | Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title_full_unstemmed | Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title_short | Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for $(K,K′)$-quasiconformal harmonic mappings of unit disk |
| title_sort | hyperbolically lipschitz continuity, area distortion and coefficient estimates for $(k,k′)$-quasiconformal harmonic mappings of unit disk |
| topic_facet | (K K′ )-quasiconformal mapping hyperbolically Lipschitz continuous area distortion coefficient estimate (K K′ )-quasiconformal mapping hyperbolically Lipschitz continuous area distortion coefficient estimate |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6041 |
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