Univalence criteria and quasiconformal extensions
We establish more general conditions for the univalence of analytic functions in the open unit disk $U$. In addition, we obtain a refinement to the criterion of quasiconformal extension for the main result.
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| author | Çaǧlar, M. Orhan, H. Саглар, М. Орхан, Х. |
| author_facet | Çaǧlar, M. Orhan, H. Саглар, М. Орхан, Х. |
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| description | We establish more general conditions for the univalence of analytic functions in the open unit disk $U$. In addition, we obtain a refinement to the criterion of quasiconformal extension for the main result. |
| first_indexed | 2026-03-24T02:15:04Z |
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UDC 517.5
M. Çağlar (Kafkas Univ., Kars, Turkey),
H. Orhan (Ataturk Univ., Erzurum, Turkey)
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSIONS*
КРИТЕРIЇ УНIВАЛЕНТНОСТI ТА КВАЗIКОНФОРМНI РОЗШИРЕННЯ
We establish more general conditions for the univalence of analytic functions in the open unit disk \scrU . In addition, we
obtain a refinement to the criterion of quasiconformal extension for the main result.
Встановлено бiльш загальнi умови унiвалентностi аналiтичних функцiй у вiдкритому одиничному колi \scrU . Крiм
того, уточнено критерiй квазiконформного розширення для основного результату.
1. Introduction. Let \scrA be the class of analytic functions f in the open unit disk \scrU = \{ z \in \BbbC : | z| < 1\}
with f(0) = f \prime (0) - 1 = 0. We denote by \scrU r the disk \{ z \in \BbbC : | z| < r\} , where 0 < r \leq 1, by
\scrU = \scrU 1 the open unit disk of the complex plane and by I the interval [0,\infty ).
Most important and known univalence criteria for analytic functions defined in the open unit disk
were obtained by Becker [3], Nehari [18] and Ozaki, Nunokawa [19]. Some extensions of these three
criteria were given by (see [15, 17, 20, 24 – 27, 29]). During the time, a lot of univalence criteria
were obtained by different authors (see also [7, 9 – 11]).
In the present investigation we use the method of subordination chains to establish some sufficient
conditions for the univalence of an analytic function. Also, by using Becker’s method, we obtain a
refinement to the criterion of quasiconformal extension for the main result.
2. Preliminaries. Before proving our main theorem we need a brief summary of the method of
Loewner chains and quasiconformal extensions.
A function L(z, t) : \scrU \times [0,\infty ) \rightarrow \BbbC is said to be subordination chain (or Loewner chain) if:
(i) L(z, t) is analytic and univalent in \scrU for all t \geq 0.
(ii) L(z, t) \prec L(z, s) for all 0 \leq t \leq s < \infty , where the symbol “\prec ” stands for subordination.
In proving our results, we will need the following theorem due to Pommerenke [23].
Theorem 2.1. Let L(z, t) = a1(t)z + a2(t)z
2 + . . . , a1(t) \not = 0 be analytic in \scrU r for all t \in I,
locally absolutely continuous in I, and locally uniform with respect to \scrU r . For almost all t \in I,
suppose that
z
\partial L(z, t)
\partial z
= p(z, t)
\partial L(z, t)
\partial t
\forall z \in \scrU r, (2.1)
where p(z, t) is analytic in \scrU and satisfies the condition \Re p(z, t) > 0 for all z \in \scrU , t \in I. If
| a1(t)| \rightarrow \infty for t \rightarrow \infty and \{ L(z, t)\diagup a1(t)\} forms a normal family in \scrU r, then, for each t \in I,
the function L(z, t) has an analytic and univalent extension to the whole disk \scrU .
Let k be constant in [0, 1). Then a homeomorphism f of G \subset \BbbC is said to be k-quasiconformal,
if \partial zf and \partial zf in the distributional sense are locally integrable on G and fulfill the inequality
| \partial zf | \leq k | \partial zf | almost everywhere in G. If we do not need to specify k, we will simply call that f
is quasiconformal.
The method of constructing quasiconformal extension criteria is based on the following result due
to Becker (see [3, 4] and also [5]).
* This paper was supported by Atatürk University Rectorship under “The Scientific and Research Project of Atatürk
University” (Project No 2012/173).
c\bigcirc M. ÇAĞLAR, H. ORHAN, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8 1147
1148 M. ÇAĞLAR, H. ORHAN
Theorem 2.2. Suppose that L(z, t) is a Loewner chain. Consider
w(z, t) =
p(z, t) - 1
p(z, t) + 1
, z \in \scrU , t \geq 0,
where p(z, t) is given in (2.1). If
| w(z, t)| \leq k, 0 \leq k < 1,
for all z \in \scrU and t \geq 0, then L(z, t) admits a continuous extension to \scrU for each t \geq 0 and the
function F (z, z) defined by
F (z, z) =
\left\{
\scrL (z, 0), if | z| < 1,
\scrL
\biggl(
z
| z|
, \mathrm{l}\mathrm{o}\mathrm{g} | z|
\biggr)
, if | z| \geq 1,
is a k-quasiconformal extension of L(z, 0) to \BbbC .
Examples of quasiconformal extension criteria can be found in [1, 2, 6, 16, 22] and more recently
in [8, 12 – 14, 28].
3. Main results. Making use of Theorem 2.1 we can prove now, our main results.
Theorem 3.1. Consider f \in \scrA and g be an analytic function in \scrU , g(z) = 1 + b1z + . . . . Let
\alpha , \beta , A and B complex numbers such that \Re (\alpha ) >
1
2
, A + B \not = 0, | A - B| < 2, | A| \leq 1 and
| B| \leq 1. If the inequalities \bigm| \bigm| \bigm| \bigm| 1\alpha
\biggl(
f \prime (z)
g(z) - \beta
- 1
\biggr) \bigm| \bigm| \bigm| \bigm| < | A+B|
2 - | A - B|
(3.1)
and\bigm| \bigm| \bigm| \bigm| \bigm|
\biggl(
f \prime (z)
g(z) - \beta
- 1
\biggr)
| z| 2 +
\Bigl(
1 - | z| 2
\Bigr) \biggl[ \biggl( 1 - \alpha
\alpha
\biggr)
zf \prime (z)
f(z)
+
zg\prime (z)
g(z) - \beta
\biggr]
-
\bigl(
A - B
\bigr)
(A+B)
4 - | A - B| 2
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2 | A+B|
4 - | A - B| 2
(3.2)
are satisfied for all z \in \scrU , then the function f is univalent in \scrU .
Proof. We prove that there exists a real number r \in (0, 1] such that the function L : \scrU r\times I \rightarrow \BbbC ,
defined formally by
L(z, t) = f1 - \alpha (e - tz)
\bigl[
f(e - tz) +
\bigl(
et - e - t
\bigr)
z
\bigl(
g(e - tz) - \beta
\bigr) \bigr] \alpha
(3.3)
is analytic in \scrU r for all t \in I.
Since f(z) \not = 0 for all z \in \scrU \diagdown \{ 0\} , the function
\varphi 1(z, t) =
\bigl(
et - e - t
\bigr)
z
\bigl(
g(e - tz) - \beta
\bigr)
f(e - tz)
is analytic in \scrU .
It follows from
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSIONS 1149
\varphi 2(z, t) = 1 +
\bigl(
et - e - t
\bigr)
z
\bigl(
g(e - tz) - \beta
\bigr)
f(e - tz)
that there exist a r1, 0 < r1 < r such that \varphi 2 is analytic in \scrU r1 and \varphi 2(0, t) = (1 - \beta ) e2t + \beta ,
\varphi 2(z, t) \not = 0 for all z \in \scrU r1 , t \in I. Therefore, we choose an analytic branch in \scrU r1 of the function
\varphi 3(z, t) = [\varphi 2(z, t)]
\alpha .
From these considerations it follows that the function
L(z, t) = f1 - \alpha (e - tz)
\bigl[
f(e - tz) +
\bigl(
et - e - t
\bigr)
z
\bigl(
g(e - tz) - \beta
\bigr) \bigr] \alpha
=
= f(e - tz)\varphi 3(z, t) = a1(t)z + . . .
is an analytic function in \scrU r1 for all t \in I.
After simple calculation we have
a1(t) = e(2\alpha - 1)t
\bigl[
\beta e - 2t + 1 - \beta
\bigr] \alpha
for which we consider the uniform branch equal to 1 at the origin. Because \Re (\alpha ) > 1
2
, we have that
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
| a1(t)| = \infty .
Moreover, a1(t) \not = 0 for all t \in I.
From the analyticity of L(z, t) in \scrU r1 , it follows that there exists a number r2, 0 < r2 < r1, and
a constant K = K(r2) such that\bigm| \bigm| \bigm| \bigm| L(z, t)a1(t)
\bigm| \bigm| \bigm| \bigm| < K \forall z \in \scrU r2 , t \in I.
Then, by Montel’s theorem,
\biggl\{
L(z, t)
a1(t)
\biggr\}
t\in I
is a normal family in \scrU r2 . From the analyticity of
\partial L(z, t)
\partial t
,
we obtain that for all fixed numbers T > 0 and r3, 0 < r3 < r2, there exists a constant K1 > 0 (that
depends on T and r3) such that\bigm| \bigm| \bigm| \bigm| \partial L(z, t)\partial t
\bigm| \bigm| \bigm| \bigm| < K1 \forall z \in \scrU r3 , t \in [0, T ] .
Therefore, the function L(z, t) is locally absolutely continuous in I, locally uniform with respect
to \scrU r3 .
Let p : \scrU r \times I \rightarrow \BbbC be the analytic function in \scrU r, 0 < r < r3, for all t \in I, defined by
p(z, t) =
\partial L(z, t)
\partial t
\diagup z
\partial L(z, t)
\partial z
.
If the function
w(z, t) =
p(z, t) - 1
A+Bp(z, t)
=
\partial L(z, t)
\partial t
- z\partial L(z, t)
\partial z
A
z\partial L(z, t)
\partial z
+B
\partial L(z, t)
\partial t
(3.4)
is analytic in \scrU \times I and | w(z, t)| < 1, for all z \in \scrU and t \in I, then p(z, t) has an analytic extension
with positive real part in \scrU , for all t \in I. From equality (3.4) we obtain
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1150 M. ÇAĞLAR, H. ORHAN
w(z, t) =
- 2\phi (z, t)
(A - B)\phi (z, t) +A+B
(3.5)
for z \in \scrU and t \in I, where
\phi (z, t) =
\biggl(
1
\alpha
f \prime (e - tz)
g(e - tz) - \beta
- 1
\biggr)
e - 2t +
\bigl(
1 - e - 2t
\bigr) \biggl[ \biggl( 1 - \alpha
\alpha
\biggr)
e - tzf \prime (e - tz)
f(e - tz)
+
e - tzg\prime (e - tz)
g(e - tz) - \beta
\biggr]
.
(3.6)
From (3.1), (3.5), (3.6) and \Re (\alpha ) > 1
2
we have
| w(z, 0)| =
\bigm| \bigm| \bigm| \bigm| 1\alpha
\biggl(
f \prime (z)
g(z) - \beta
- 1
\biggr) \bigm| \bigm| \bigm| \bigm| < | A+B|
2 - | A - B|
and
| w(0, t)| =
\bigm| \bigm| \bigm| \bigm| \biggl( 1
\alpha (1 - \beta )
- 1
\biggr)
e - 2t
\bigm| \bigm| \bigm| \bigm| < | A+B|
2 - | A - B|
,
where A+B \not = 0, | A - B| < 2, | A| \leq 1 and | B| \leq 1.
Since
\bigm| \bigm| e - tz
\bigm| \bigm| \leq \bigm| \bigm| e - t
\bigm| \bigm| = e - t < 1 for all z \in \scrU = \{ z \in \BbbC : | z| \leq 1\} and t > 0, we find that
w(z, t) is an analytic function in \scrU . Using the maximum modulus principle it follows that for all
z \in \scrU - \{ 0\} and each t > 0 arbitrarily fixed there exists \theta = \theta (t) \in \BbbR such that
| w(z, t)| < \mathrm{m}\mathrm{a}\mathrm{x}
| z| =1
| w(z, t)| =
\bigm| \bigm| \bigm| w(ei\theta , t)\bigm| \bigm| \bigm|
for all z \in \scrU and t \in I.
Denote u = e - tei\theta . Then | u| = e - t and from (3.5) we get\bigm| \bigm| \bigm| w(ei\theta , t)\bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| 2\phi (ei\theta , t)
(A - B)\phi (ei\theta , t) +A+B
\bigm| \bigm| \bigm| \bigm| ,
where
\phi (ei\theta , t) =
\biggl(
1
\alpha
f \prime (u)
g(u) - \beta
- 1
\biggr)
| u| 2 +
\Bigl(
1 - | u| 2
\Bigr) \biggl[ \biggl(
1 - \alpha
\alpha
\biggr)
uf \prime (u)
f(u)
+
ug\prime (u)
g(u) - \beta
\biggr]
.
Because u \in \scrU , the inequality (3.2) implies that\bigm| \bigm| \bigm| w(ei\theta , t)\bigm| \bigm| \bigm| \leq 1
for all z \in \scrU and t \in I. Therefore | w(z, t)| < 1 for all z \in \scrU and t \in I.
Since all the conditions of Theorem 2.1 are satisfied, we obtain that the function L(z, t) has
an analytic and univalent extension to the whole unit disk \scrU , for all t \in I. For t = 0 we have
L(z, 0) = f(z), for z \in \scrU and therefore the function f is analytic and univalent in \scrU .
Theorem 3.1 is proved.
Remark 3.1. Some particular cases of Theorem 3.1 are the following:
(i) When \alpha = 1, \beta = 0, A = B = 1 and g(z) = f \prime (z) inequality (3.2) becomes\Bigl(
1 - | z| 2
\Bigr) \bigm| \bigm| \bigm| \bigm| zf \prime \prime (z)
f \prime (z)
\bigm| \bigm| \bigm| \bigm| \leq 1, z \in \scrU , (3.7)
which is Becker’s condition of univalence [3].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSIONS 1151
(ii) A result due to N. N. Pascu [21] is obtained when \alpha = 1, A = B = 1 and g(z) = f \prime (z).
Remark 3.2. It is worth to notice that the condition (3.2) assures the univalence of an analytic
function in more general case than that of condition (3.7).
Remark 3.3. If we put g(z) =
f(z)
z
, \alpha = 1 and \beta = 0 into Theorem 3.1, we have\bigm| \bigm| \bigm| \bigm| zf \prime (z)
f(z)
- 1
\bigm| \bigm| \bigm| \bigm| \leq | A+B|
2 - | A - B|
, z \in \scrU ,
the class of functions starlike with respect to origin.
4. Quasiconformal extension. In this section we will obtain the univalence condition given in
Theorem 3.1 to a quasiconformal extension criterion.
Theorem 4.1. Consider f \in \scrA , g be an analytic function in \scrU , g(z) = 1 + b1z + . . . and
k \in [0, 1) . Let \alpha , \beta , A and B complex numbers such that \Re (\alpha ) > 1
2
, A+B \not = 0, k | A - B| < 2,
| A| \leq 1 and | B| \leq 1. If the inequalities\bigm| \bigm| \bigm| \bigm| 1\alpha
\biggl(
f \prime (z)
g(z) - \beta
- 1
\biggr) \bigm| \bigm| \bigm| \bigm| < k | A+B|
2 - k | A - B|
and\bigm| \bigm| \bigm| \bigm| \bigm|
\biggl(
f \prime (z)
g(z) - \beta
- 1
\biggr)
| z| 2 +
\Bigl(
1 - | z| 2
\Bigr) \biggl[ \biggl(
1 - \alpha
\alpha
\biggr)
zf \prime (z)
f(z)
+
zg\prime (z)
g(z) - \beta
\biggr]
-
k2
\bigl(
A - B
\bigr)
(A+B)
4 - k2 | A - B| 2
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 2k | A+B|
4 - k2 | A - B| 2
are satisfied for all z \in \scrU , then the function f has a k-quasiconformal extension to \BbbC .
Proof. In the proof of Theorem 3.1 has been proved that the function L(z, t) given by (3.3) is
a subordination chain in \scrU . Applying Theorem 2.2 to the function w(z, t) given by (3.5), we obtain
that the assumption
| w(z, t)| =
\bigm| \bigm| \bigm| \bigm| - 2\phi (z, t)
(A - B)\phi (z, t) +A+B
\bigm| \bigm| \bigm| \bigm| \leq k, z \in \scrU , t \geq 0, k \in [0, 1), (4.1)
where \phi (z, t) is defined by (3.6).
Lenghty but elementary calculation shows that the last inequality (4.1) is equivalent to\bigm| \bigm| \bigm| \bigm| \biggl( 1
\alpha
f \prime (e - tz)
g(e - tz) - \beta
- 1
\biggr)
e - 2t +
\bigl(
1 - e - 2t
\bigr) \biggl[ \biggl( 1 - \alpha
\alpha
\biggr)
e - tzf \prime (e - tz)
f(e - tz)
+
e - tzg\prime (e - tz)
g(e - tz) - \beta
\biggr]
-
-
k2
\bigl(
A - B
\bigr)
(A+B)
4 - k2 | A - B| 2
\bigm| \bigm| \bigm| \bigm| \bigm| \leq 2k | A+B|
4 - k2 | A - B| 2
. (4.2)
Inequality (4.2) implies k-quasiconformal extensibility of f.
Theorem 4.1 is proved.
Remark 4.1. For A = B = 1 \alpha = 1, \beta = 0, g = f \prime in Theorem 4.1, we have result of
Becker [3].
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
1152 M. ÇAĞLAR, H. ORHAN
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Received 15.02.13,
after revision — 01.07.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 8
|
| id | umjimathkievua-article-1910 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:04Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fa/f900b27fe30df45b4aaca0906c7e02fa.pdf |
| spelling | umjimathkievua-article-19102019-12-05T09:31:14Z Univalence criteria and quasiconformal extensions Критерiї унiвалентностi та квазiконформнi розширення Çaǧlar, M. Orhan, H. Саглар, М. Орхан, Х. We establish more general conditions for the univalence of analytic functions in the open unit disk $U$. In addition, we obtain a refinement to the criterion of quasiconformal extension for the main result. Встановлено бiльш загальнi умови унiвалентностi аналiтичних функцiй у вiдкритому одиничному колi $U$. Крiм того, уточнено критерiй квазiконформного розширення для основного результату. Institute of Mathematics, NAS of Ukraine 2016-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1910 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 8 (2016); 1147-1151 Український математичний журнал; Том 68 № 8 (2016); 1147-1151 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1910/892 Copyright (c) 2016 Çaǧlar M.; Orhan H. |
| spellingShingle | Çaǧlar, M. Orhan, H. Саглар, М. Орхан, Х. Univalence criteria and quasiconformal extensions |
| title | Univalence criteria and quasiconformal extensions |
| title_alt | Критерiї унiвалентностi та квазiконформнi розширення |
| title_full | Univalence criteria and quasiconformal extensions |
| title_fullStr | Univalence criteria and quasiconformal extensions |
| title_full_unstemmed | Univalence criteria and quasiconformal extensions |
| title_short | Univalence criteria and quasiconformal extensions |
| title_sort | univalence criteria and quasiconformal extensions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1910 |
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