Univalence criteria and quasiconformal extension of a general integral operator
UDC 517.5 We give some sufficient conditions of analyticity and univalence for functions defined by an integral operator. Next, we refine the result to a quasiconformal extension criterion with the help of the Becker’s method. Further, new univalence criteria and the significant relationships with o...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507159430692864 |
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| author | Deniz, E. Kanas, S. Orhan, H. Deniz, E. Kanas, S. Orhan, H. KANAS, STANIS LAWA ORHAN, HALIT |
| author_facet | Deniz, E. Kanas, S. Orhan, H. Deniz, E. Kanas, S. Orhan, H. KANAS, STANIS LAWA ORHAN, HALIT |
| author_sort | Deniz, E. |
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| datestamp_date | 2022-03-27T15:39:11Z |
| description | UDC 517.5
We give some sufficient conditions of analyticity and univalence for functions defined by an integral operator. Next, we refine the result to a quasiconformal extension criterion with the help of the Becker’s method. Further, new univalence criteria and the significant relationships with other results are given. A number of known univalence conditions would follow upon specializing the parameters involved in main results.
  |
| doi_str_mv | 10.37863/umzh.v74i1.1148 |
| first_indexed | 2026-03-24T02:04:53Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i1.1148
UDC 517.5
E. Deniz (Kafkas Univ., Kars, Turkey),
S. Kanas (Univ. Rzeszów, Poland),
H. Orhan (Atatürk Univ., Erzurum, Turkey)
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSION
OF A GENERAL INTEGRAL OPERATOR*
УНIВАЛЕНТНI КРИТЕРIЇ ТА КВАЗIКОНФОРМНЕ РОЗШИРЕННЯ
IНТЕГРАЛЬНОГО ОПЕРАТОРА ЗАГАЛЬНОГО ВИГЛЯДУ
We give some sufficient conditions of analyticity and univalence for functions defined by an integral operator. Next, we
refine the result to a quasiconformal extension criterion with the help of the Becker’s method. Further, new univalence
criteria and the significant relationships with other results are given. A number of known univalence conditions would
follow upon specializing the parameters involved in main results.
Запропоновано достатнi умови аналiтичностi та унiвалентностi для функцiй, що визначаються деяким iнтегральним
оператором. Цей результат зводиться до критерiю квазiконформного розширення за допомогою методу Бекера. Далi
отримано новi критерiї унiвалентностi та вказано важливi зв’язки з iншими результатами. Також з основного
результату при рiзних значеннях параметрiв, якi задiянi у формулюваннi цього результату, випливають деякi вiдомi
умови унiвалентностi.
1. Introduction. Denote by \scrU r =
\bigl\{
z \in \BbbC : | z| < r
\bigr\}
, 0 < r \leq 1, the disk of radius r and let
\scrU = \scrU 1. Let \scrA denote the class of analytic functions in the open unit disk \scrU which satisfy the
usual normalization condition f(0) = f \prime (0) - 1 = 0, and let \scrS be the subclass of \scrA consisting
of the functions f which are univalent in \scrU . Also, let \scrP denote the class of functions p(z) =
= 1 +
\sum \infty
n=1
pkz
k that satisfy the condition \Re p(z) > 0 (z \in \scrU ), and \Omega be a class of functions w
which are analytic in \scrU and such that | w(z)| < 1 for z \in \scrU . These classes have been one of the
important subjects of research in geometric function theory for a long time (see [34]).
We say that a sense-preserving homeomorphism f of a plane domain G \subset \BbbC is k-quasi-
conformal, if f is absolutely continuous on almost all lines parallel to coordinate axes and | fz| \leq
\leq k| fz| , almost everywhere in G, where fz = \partial f/\partial z, fz = \partial f/\partial z and k is a real constant with
0 \leq k < 1. For the general defnition of quasiconformal mappings see [1].
Univalence of complex functions is an important property but, in many cases is impossible
to show directly that a certain function is univalent. For this reason, many authors found differ-
ent sufficient conditions of univalence. Two of the most important are the well-known criteria of
Becker [3] and Ahlfors [1]. Becker and Ahlfors’ works depend upon a ingenious use of the the-
ory of the Loewner chains and the generalized Loewner differential equation. Extensions of these
two criteria were given by Ruscheweyh [30], Singh and Chichra [33], Kanas and Lecko [14, 15]
and Lewandowski [18]. The recent investigations on this subject are due to Raducanu et al. [29]
and Deniz and Orhan [8, 9]. Furthermore, Pascu [24] and Pescar [25] obtained some extensions of
Becker’s univalence criterion for an integral operator, while Ovesea [23] obtained a generalization of
Ruscheweyh’s univalence criterion for an integral operator.
* This paper was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering
Knowledge, Faculty of Mathematics and Natural Sciences, University of Rzeszów.
c\bigcirc E. DENIZ, S. KANAS, H. ORHAN, 2022
24 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSION OF A GENERAL INTEGRAL . . . 25
In the present paper, we formulate a new criteria for univalence of the functions defined by
an integral operator G\alpha , considered in [23], and improve obtained there results. Also, we obtain
a refinement to a quasiconformal extension criterion of the main result. In the special cases, our
univalence conditions contain the results obtained by some of the authors cited in references. Our
considerations are based on the theory of Loewner chains.
2. Loewner chains and quasiconformal extension. The method of Loewner chains will prove
to be crucial in our later consideration therefore we present a brief summary of that method.
Let \scrL (z, t) = a1(t)z+a2(t)z
2+. . . , a1(t) \not = 0, be a function defined on \scrU \times I, where I := [0,\infty )
and a1(t) is a complex-valued, locally absolutely continuous function on I. Then \scrL (z, t) is said to
be Loewner chain if \scrL (z, t) has the following conditions:
(i) \scrL (z, t) is analytic and univalent in \scrU for all t \in I;
(ii) \scrL (z, t) \prec \scrL (z, s) for all 0 \leq t \leq s < \infty ,
where the symbol \prec stands for subordination. If a1(t) = et, then we say that \scrL (z, t) is a standard
Loewner chain.
In order to prove main results we need the following theorem due to Pommerenke [27] (see also
[28]). This theorem is often used to find out univalency for an analytic function, apart from the
theory of Loewner chains.
Theorem 2.1 [28]. Let \scrL (z, t) = a1(t)z+a2(t)z
2+ . . . be analytic in \scrU r for all t \in I. Suppose
that:
(i) \scrL (z, t) is a locally absolutely continuous function in the interval I, and locally uniformly
with respect to \scrU r;
(ii) a1(t) is a complex valued continuous function on I such that a1(t) \not = 0, | a1(t)| \rightarrow \infty for
t \rightarrow \infty and \biggl\{
\scrL (z, t)
a1(t)
\biggr\}
t\in I
forms a normal family of functions in \scrU r;
(iii) there exists an analytic function p : \scrU \times I \rightarrow \BbbC satisfying \Re p(z, t) > 0 for all z \in \scrU , t \in I
and
z
\partial \scrL (z, t)
\partial z
= p(z, t)
\partial \scrL (z, t)
\partial t
, z \in \scrU r, t \in I. (2.1)
Then, for each t \in I, \scrL (\cdot , t)has an analytic and univalent extension to the whole disk \scrU and \scrL (z, t)
is a Loewner chain.
The equation (2.1) is called the generalized Loewner differential equation.
The following strengthening of Theorem 2.1 leads to the method of constructing quasiconformal
extension, and is based on the result due to Becker (see [3 – 5]).
Theorem 2.2 [3 – 5]. Suppose that \scrL (z, t) is a Loewner chain for which p(z, t), defined in (2.1),
satisfies the condition
p(z, t) \in U(k) :=
\biggl\{
w \in \BbbC :
\bigm| \bigm| \bigm| \bigm| w - 1
w + 1
\bigm| \bigm| \bigm| \bigm| \leq k
\biggr\}
=
=
\biggl\{
w \in \BbbC :
\bigm| \bigm| \bigm| \bigm| w - 1 + k2
1 - k2
\bigm| \bigm| \bigm| \bigm| \leq 2k
1 - k2
\biggr\}
, 0 \leq k < 1,
for all z \in \scrU and t \in I . Then \scrL (z, t) admits a continuous extension to \scrU for each t \in I and the
function F (z, \=z) defined by
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
26 E. DENIZ, S. KANAS, H. ORHAN
F (z, \=z) =
\left\{
\scrL (z, 0) \itf \ito \itr | z| < 1,
\scrL
\biggl(
z
| z|
, \mathrm{l}\mathrm{o}\mathrm{g} | z|
\biggr)
\itf \ito \itr | z| \geq 1,
is a k-quasiconformal extension of \scrL (z, 0) to \BbbC .
Detailed information about Loewner chains and quasiconformal extension criterion can be found
in [1, 2, 6, 7, 17, 26]. For a recent account of the theory we refer the reader to [12, 13]. One can also
see the following studies [10, 11, 16, 31, 32] dealing with local and boundary behavior of conformal,
quasiconformal and quasiregular mappings, as well as their generalizations.
3. Univalence criteria. The first theorem is our glimpse of the role of the generalized Loewner
chains in univalence results for an operator G\alpha , studied in [23]. The theorem formulates the condi-
tions under which such an operator is analytic and univalent.
Theorem 3.1. Let \alpha , c and s be complex numbers such that c /\in [0,\infty ); s = a + ib, a > 0,
b \in \BbbR ; m > 0 and f, g \in \scrA . If there exists a function h, analytic in \scrU and such that h(0) = h0,
h0 \in \BbbC , h0 /\in ( - \infty , 0], and the inequalities\bigm| \bigm| \bigm| \alpha - m
2a
\bigm| \bigm| \bigm| < m
2a
, (3.1)\bigm| \bigm| \bigm| \bigm| c
h(z)
+
m
2\alpha
\bigm| \bigm| \bigm| \bigm| < m
2 | \alpha |
, (3.2)
and \bigm| \bigm| \bigm| \bigm| - c\alpha
ah(z)
| z| m/a +
\Bigl(
1 - | z| m/a
\Bigr) \biggl[
(\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
\biggr]
- m
2a
\bigm| \bigm| \bigm| \bigm| \leq m
2a
(3.3)
hold true for all z \in \scrU , then the function
G\alpha (z) =
\left[ \alpha z\int
0
g\alpha - 1(u)f \prime (u)du
\right] 1/\alpha
(3.4)
is analytic and univalent in \scrU , where the principal branch is intended.
Proof. We first note that G\alpha is well defined and analytic in the unit disk. We rewrite G\alpha in the
form
G\alpha (z) =
\left[ \alpha z\int
0
u\alpha - 1
\biggl(
g(u)
u
\biggr) \alpha - 1
f \prime (u)du
\right] 1/\alpha
,
where singularity of g(z)/z at z = 0 is removed. Because g \in \scrA , the function g(z)/z = 1 + . . . is
analytic in \scrU , and then there exists a disc \scrU r1 , 0 < r1 \leq 1, in which g(z)/z \not = 0 for all z \in \scrU r1 .
By changing the variable, we next have
G\alpha (z) = z
\left[ \alpha 1\int
0
w\alpha - 1
\biggl(
g(zw)
zw
\biggr) \alpha - 1
f \prime (zw)dw
\right] 1/\alpha
= z + c2z
2 + . . . ,
so that G\alpha is analytic in some neighbourhood of the origin.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSION OF A GENERAL INTEGRAL . . . 27
Next, we prove that there exists a real number r \in (0, 1] such that the function \scrL (\cdot , t), defined
formally by
\scrL (z, t) =
=
\left( \alpha e - stz\int
0
g\alpha - 1(u)f \prime (u)du - a
c
\bigl(
emt - 1
\bigr)
e - stzg\alpha - 1(e - stz)f \prime (e - stz)h(e - stz)
\right)
1/\alpha
, (3.5)
is analytic in \scrU r for all t \in [0,\infty ) = I.
Denoting \phi (z) = g(z)/z we define now a function
\phi 1(z, t) = \alpha
e - stz\int
0
u\alpha - 1\phi (u)f \prime (u)du= e - st\alpha z\alpha + . . . ,
so that \phi 1 can be rewritten in the form
\phi 1(z, t) = z\alpha \phi 2(z, t),
where \phi 2 is analytic in \scrU r1 . Hence, the function
\phi 3(z, t) = \phi 2(z, t) -
a
c
\bigl(
emt - 1
\bigr)
e - st\alpha \phi (e - stz)f \prime (e - stz)h(e - stz)
is analytic in \scrU r1 and
\phi 3(0, t) = e - st\alpha
\Bigl[ \Bigl(
1 +
a
c
h0
\Bigr)
- a
c
h0e
mt
\Bigr]
.
Now, we prove that \phi 3(0, t) \not = 0 for all t \in I. It is easy to see that \phi 3(0, 0) = 1. Suppose that
there exists t0 > 0 such that \phi 3(0, t0) = 0. Then the equality emt0 =
c+ ah0
ah0
holds. Since
h0 /\in ( - \infty , 0], this equality implies that c > 0, which contradicts c /\in [0,\infty ). From this we conclude
that \phi 3(0, t) \not = 0 for all t \in I. Therefore, there is a disk \scrU r2 , r2 \in (0, r1], in which \phi 3(z, t) \not = 0
for all t \in I. Thus, we can choose a principal branch of
\bigl[
\phi 3(z, t)
\bigr] 1/\alpha
analytic in \scrU r2 . By the
construction of \scrL (z, t) and (3.5) we have that
\scrL (z, t) = z
\bigl[
\phi 3(z, t)
\bigr] 1/\alpha
= a1(t)z + a2(t)z
2 + . . .
and, consequently, the function \scrL (z, t) is analytic in \scrU r2 .
We note that
a1(t) = et(
m
\alpha
- s)
\Bigl[ \Bigl(
1 +
a
c
h0
\Bigr)
e - mt - a
c
h0
\Bigr] 1/\alpha
,
for which we consider the principal branch equal to 1 at the origin.
Since
\bigm| \bigm| \bigm| a\alpha - m
2
\bigm| \bigm| \bigm| < m
2
is equivalent to \Re
\Bigl\{ m
\alpha
\Bigr\}
> a = \Re (s), we get
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
\bigm| \bigm| a1(t)\bigm| \bigm| = \infty .
Moreover, a1(t) \not = 0 for all t \in I.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
28 E. DENIZ, S. KANAS, H. ORHAN
From the analyticity of \scrL (z, t) in \scrU r2 , it follows that there exist a number r3 such that 0 < r3 <
< r2 and a constant K = K(r3) such that\bigm| \bigm| \bigm| \bigm| \scrL (z, t)a1(t)
\bigm| \bigm| \bigm| \bigm| < K, z \in \scrU r3 , t \in I.
By the Montel’s theorem [22],
\biggl\{
\scrL (z, t)
a1(t)
\biggr\}
t\in I
forms a normal family in \scrU r3 . From the analyticity of
\partial \scrL (z, t)
\partial t
, it may be concluded that for all fixed numbers T > 0 and r4, 0 < r4 < r3, there exists a
constant K1 > 0 (that depends on T and r4) such that\bigm| \bigm| \bigm| \bigm| \partial \scrL (z, t)\partial t
\bigm| \bigm| \bigm| \bigm| < K1, z \in \scrU r4 , t \in [0, T ].
Therefore, the function \scrL (z, t) is locally absolutely continuous in I, locally uniform with respect
to \scrU r4 .
Let p : \scrU r \times I \rightarrow \BbbC denote a function
p(z, t) = z
\partial \scrL (z, t)
\partial z
\bigg/
\partial \scrL (z, t)
\partial t
,
that is, analytic in \scrU r, 0 < r < r4, for all t \in I (the singularity at z = 0 is removable). If the
function
w(z, t) =
p(z, t) - 1
p(z, t) + 1
=
z\partial \scrL (z, t)
\partial z
- \partial \scrL (z, t)
\partial t
z\partial \scrL (z, t)
\partial z
+
\partial \scrL (z, t)
\partial t
(3.6)
is analytic in \scrU \times I and
\bigm| \bigm| w(z, t)\bigm| \bigm| < 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. According to (3.6), we have
w(z, t) =
(1 + s)A(z, t) - m
(1 - s)A(z, t) +m
, (3.7)
where
A(z, t) =
- c\alpha
ah(e - stz)
e - mt +
\bigl(
1 - e - mt
\bigr) \biggl[
(\alpha - 1)
e - stzg\prime (e - stz)
g(e - stz)
+
+1 +
e - stzf \prime \prime (e - stz)
f \prime (e - stz)
+
e - stzh\prime (e - stz)
h(e - stz)
\biggr]
(3.8)
for z \in \scrU and t \in I. Hence, the inequality
\bigm| \bigm| w(z, t)\bigm| \bigm| < 1 is equivalent to\bigm| \bigm| \bigm| A(z, t) - m
2a
\bigm| \bigm| \bigm| < m
2a
, a = \Re (s), z \in \scrU , t \in I.
Define now
B(z, t) = A(z, t) - m
2a
, z \in \scrU , t \in I.
From (3.1), (3.2) and (3.8) it follows that
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSION OF A GENERAL INTEGRAL . . . 29
\bigm| \bigm| B(z, 0)
\bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| c\alpha
ah(z)
+
m
2a
\bigm| \bigm| \bigm| \bigm| < m
2a
(3.9)
and
\bigm| \bigm| B(0, t)
\bigm| \bigm| = 1
a
\bigm| \bigm| \bigm| \bigm| c\alpha e - mt
h0
- a\alpha
\bigl(
1 - e - mt
\bigr)
+
m
2
\bigm| \bigm| \bigm| \bigm| =
=
1
a
\bigm| \bigm| \bigm| \bigm| \biggl( c\alpha
h0
+
m
2
\biggr)
e - mt +
\Bigl( m
2
- a\alpha
\Bigr) \bigl(
1 - e - mt
\bigr) \bigm| \bigm| \bigm| \bigm| < m
2a
. (3.10)
Since | e - stz| \leq | e - st| = e - at < 1 for all z \in \scrU =
\bigl\{
z \in \BbbC : | z| \leq 1
\bigr\}
and t > 0, we conclude that
for each t > 0 B(z, t) is an analytic function in \scrU . Using the maximum modulus principle it follows
that for all z \in \scrU \setminus \{ 0\} and each t > 0 arbitrarily fixed there exists \theta = \theta (t) \in \BbbR such that\bigm| \bigm| B(z, t)
\bigm| \bigm| < \mathrm{l}\mathrm{i}\mathrm{m}
| z| =1
\bigm| \bigm| B(z, t)
\bigm| \bigm| = \bigm| \bigm| \bigm| B(ei\theta , t)
\bigm| \bigm| \bigm| (3.11)
for all z \in \scrU and t \in I.
Denote u = e - stei\theta . Then | u| = e - at, and from (3.8) we obtain\bigm| \bigm| \bigm| B(ei\theta , t)
\bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| c\alpha
ah(u)
| u| m/a +
m
2a
-
\Bigl(
1 - | u| m/a
\Bigr) \biggl[
(\alpha - 1)
ug\prime (u)
g(u)
+1 +
uf \prime \prime (u)
f \prime (u)
+
uh\prime (u)
h(u)
\biggr] \bigm| \bigm| \bigm| \bigm| .
Since u \in \scrU , the inequality (3.3) implies that\bigm| \bigm| \bigm| B(ei\theta , t)
\bigm| \bigm| \bigm| \leq m
2a
,
and from (3.9), (3.10) and (3.11), we conclude that\bigm| \bigm| B(z, t)
\bigm| \bigm| = \bigm| \bigm| \bigm| A(z, t) - m
2a
\bigm| \bigm| \bigm| < m
2a
for all z \in \scrU and t \in I. Therefore,
\bigm| \bigm| w(z, t)\bigm| \bigm| < 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 \scrL (z, t) has
an analytic and univalent extension to the whole unit disk \scrU for all t \in I. For t = 0, we have
\scrL (z, 0) = G\alpha (z) for z \in \scrU and, therefore, the function G\alpha (z) is analytic and univalent in \scrU .
Theorem 3.1 is proved.
Abbreviating (3.3), we can now rephrase Theorem 3.1 in a simpler form.
Theorem 3.2. Let f, g \in \scrA . Let m > 0, the complex numbers \alpha , c, s and the function h be as
in Theorem 3.1. Moreover, suppose that the inequalities (3.1) and (3.2) are satisfied. If the inequality\bigm| \bigm| \bigm| \bigm| (\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
- m
2a
\bigm| \bigm| \bigm| \bigm| \leq m
2a
(3.12)
holds true for all z \in \scrU , then the function G\alpha defined by (3.4) is analytic and univalent in \scrU .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
30 E. DENIZ, S. KANAS, H. ORHAN
Proof. Making use of (3.2) and (3.12), we obtain\bigm| \bigm| \bigm| \bigm| c\alpha
h(z)
| z| m/a +
m
2
- a
\Bigl(
1 - | z| m/a
\Bigr) \biggl[
(\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
\biggr] \bigm| \bigm| \bigm| \bigm| =
=
\bigm| \bigm| \bigm| \bigm| \biggl( c\alpha
h(z)
+
m
2
\biggr)
| z| m/a+
+
\Bigl(
1 - | z| m/a
\Bigr) \biggl[
- a
\biggl(
(\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
\biggr)
+
m
2
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq | z| m/am
2
+
\Bigl(
1 - | z| m/a
\Bigr) m
2
=
m
2
,
so that the condition (3.3) is satisfied. This finishes the proof, since all the assumption of Theorem 3.1
are satisfied.
The special case of Theorem 3.1, i.e., for s = \alpha = 1 and h(z) = - c, leads to the following
result.
Corollary 3.1. Let f \in \scrA and m > 1. If\bigm| \bigm| \bigm| \bigm| m - 2
2
- (1 - | z| m)
zf \prime \prime (z)
f \prime (z)
\bigm| \bigm| \bigm| \bigm| \leq m
2
holds for z \in \scrU , then the function f univalent in \scrU .
Corollary 3.1 in turn implies the well-known Becker’s univalence citerion [3].
Remark 3.1. Important examples of univalence criteria may be obtained by a suitable choices of
f and g, below.
(1) Choose g1(z) = z . Then Theorem 3.1 gives analyticity and univalence of the operator
F (z) =
\left[ \alpha z\int
0
u\alpha - 1f \prime (u)du
\right] 1/\alpha
,
which was studied by Pascu [24].
(2) Setting f(z) = z in Theorem 3.1, we obtain that the operator
G(z) =
\left[ \alpha z\int
0
g\alpha - 1(u)du
\right] 1/\alpha
is analytic and univalent in \scrU . The operator G was introduced by Moldoveanu and Pascu [20].
(3) Taking f \prime (z) =
g(z)
z
in Theorem 3.1, we find that
H(z) =
\left[ \alpha z\int
0
g\alpha (u)
u
du
\right] 1/\alpha
is analytic and univalent in \scrU . The operator H was introduced and studied by Mocanu [19].
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSION OF A GENERAL INTEGRAL . . . 31
If we limit a range of parameter a to the case a \geq 1, then, applying the Theorem 3.1, we obtain
the following theorem.
Theorem 3.3. Let \alpha , c and s be complex numbers such that c /\in [0,\infty ); s = a + ib, a \geq 1,
b \in \BbbR , m > 0 and f, g \in \scrA . Let the function h be as in Theorem 3.1. Moreover, suppose that the
inequalities (3.1) and (3.2) are satisfied. If the inequality\bigm| \bigm| \bigm| \bigm| - c\alpha
ah(z)
| z| m + (1 - | z| m)
\biggl[
(\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
\biggr]
- m
2a
\bigm| \bigm| \bigm| \bigm| \leq m
2a
(3.13)
holds true for all z \in \scrU , then the function G\alpha (z), defined by (3.4), is analytic and univalent in \scrU .
Proof. For \lambda \in [0, 1] define the linear function
\phi (z, \lambda ) = \lambda k(z) + (1 - \lambda ) l(z), z \in \scrU , t \in I,
where
k(z) =
c\alpha
h(z)
+
m
2
and
l(z) = - a
\biggl[
(\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
\biggr]
+
m
2
.
For fixed z \in \scrU and t \in I, \phi (z, \lambda ) is a point of a segment with endpoints at k(z) and l(z). The
function \phi (z, \lambda ) is analytic in \scrU for all \lambda \in [0, 1] and z \in \scrU , satisfies
| \phi (z, 1)| = | k(z)| < m
2
(3.14)
and \bigm| \bigm| \phi (z, | z| m)
\bigm| \bigm| \leq m
2
, (3.15)
which follows from (3.2) and (3.13). If \lambda increases from \lambda 1 = | z| m to \lambda 2 = 1, then the point \phi (z, \lambda )
moves on the segment whose endpoints are \phi (z, | z| m) and \phi (z, 1). Because a \geq 1, from (3.14) and
(3.15) it follows that \bigm| \bigm| \bigm| \phi (z, | z| m/a)
\bigm| \bigm| \bigm| \leq m
2
, z \in \scrU . (3.16)
We can observe that the inequality (3.16) is just the condition (3.3), and then Theorem 3.1 now yields
that the function G\alpha (z), defined by (3.4), is analytic and univalent in \scrU .
Theorem 3.3 is proved.
Remark 3.2. Applying Theorem 3.3 to m = 2 and the function h(z) \equiv 1, and g(z) = f(z),
\alpha = 1/s
\Bigl(
or g(z) = z, a = 1, c = - 1
\alpha
, respectively
\Bigr)
, we obtain the results by Ruscheweyh [30]
(or Moldoveanu and Pascu [21], respectively).
Remark 3.3. Substituting 1/h instead of h with h(0) = 1 and setting g(z) = f(z), \alpha = 1/s,
m = 2 in Theorem 3.3, we obtain the result due to Singh and Chichra [33].
Remark 3.4. Setting g(z) = f(z), s = \alpha = 1, c = - 1, m = 2 and h(z) =
k(z) + 1
2
, where
k is an analytic function with positive real part in \scrU with k(0) = 1 in Theorem 3.3, we obtain the
result by Lewandowski [18].
Remark 3.5. For the case when m = 2 and h(0) = h0 = 1 Theorems 3.1 and 3.3 reduce to the
results by Ovesea [23].
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32 E. DENIZ, S. KANAS, H. ORHAN
4. Quasiconformal extension criterion. In this section, we will refine the univalence condition
given in Theorem 3.1 to a quasiconformal extension criterion.
Theorem 4.1. Let \alpha , c and s be complex numbers such that c /\in [0,\infty ), s = a + ib, a > 0,
b \in \BbbR , m > 0; k \in [0, 1) and f, g \in \scrA . If there exists a function h, analytic in \scrU , such that
h(0) = h0, h0 \in \BbbC , h0 /\in ( - \infty , 0], and the inequalities\bigm| \bigm| \bigm| \alpha - m
2a
\bigm| \bigm| \bigm| < m
2a
,
(4.1)\bigm| \bigm| \bigm| \bigm| c\alpha
h(z)
+
m
2
\bigm| \bigm| \bigm| \bigm| < k
m
2
and \bigm| \bigm| \bigm| \bigm| - c\alpha
ah(z)
| z| m/a +
\Bigl(
1 - | z| m/a
\Bigr) \biggl[
(\alpha - 1)
zg\prime (z)
g(z)
+ 1 +
zf \prime \prime (z)
f \prime (z)
+
zh\prime (z)
h(z)
\biggr]
- m
2a
\bigm| \bigm| \bigm| \bigm| \leq k
m
2a
(4.2)
hold true for all z \in \scrU , then the function G\alpha (z) given by (3.4) has an K-quasiconformal extension
to \BbbC , where
K =
\left\{
k \itf \ito \itr s = 1,
| s - 1| 2 + k| \=s2 - 1|
| \=s2 - 1| + k| s - 1| 2
\itf \ito \itr s \not = 1.
Proof. In the proof of Theorem 3.1 it has been shown that the function \scrL (z, t), given by (3.5), is
a subordination chain in \scrU . Applying Theorem 2.2 to the function w(z, t) given by (3.7), we obtain
that the condition \bigm| \bigm| \bigm| \bigm| (1 + s)A(z, t) - m
(1 - s)A(z, t) +m
\bigm| \bigm| \bigm| \bigm| < l, z \in \scrU , t \in I, 0 \leq l < 1, (4.3)
with A(z, t) defined by (3.8), implies l-quasiconformal extensibility of G\alpha (z). Lengthy, but elemen-
tary calculations, show that inequality (4.3) is equivalent to\bigm| \bigm| \bigm| \bigm| \bigm| A(z, t) -
m
\bigl(
(1 + l2) + a(1 - l2) - ib(1 - l2)
\bigr)
2a(1 + l2) + (1 - l2)(1 + | s| 2)
\bigm| \bigm| \bigm| \bigm| \bigm| \leq 2lm
2a(1 + l2) + (1 - l2)(1 + | s| 2)
. (4.4)
Taking into account (4.1) and (4.2), we clearly see that\bigm| \bigm| \bigm| A(z, t) - m
2a
\bigm| \bigm| \bigm| \leq k
m
2a
. (4.5)
Consider the two disks \Delta 1(s1, r1) and \Delta 2(s2, r2) defined by (4.4) and (4.5), respectively, where
A(z, t) is replaced by a complex variable w. The proof is completed by showing that there exists
l \in [0, 1) for which \Delta 2 is contained in \Delta 1 . Equivalently \Delta 2 \subset \Delta 1 holds, if | s1 - s2| + r2 \leq r1,
that is,\bigm| \bigm| \bigm| \bigm| \bigm| m
\bigl(
(1 + l2) + a(1 - l2)
\bigr)
- imb(1 - l2)
2a(1 + l2) + (1 - l2)
\bigl(
1 + | s| 2
\bigr) - m
2a
\bigm| \bigm| \bigm| \bigm| \bigm| + k
m
2a
\leq 2lm
2a(1 + l2) + (1 - l2)
\bigl(
1 + | s| 2
\bigr)
or
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UNIVALENCE CRITERIA AND QUASICONFORMAL EXTENSION OF A GENERAL INTEGRAL . . . 33
(1 - l2)| \=s2 - 1|
2a
\Bigl[
2a(1 + l2) + (1 - l2)
\bigl(
1 + | s| 2
\bigr) \Bigr] \leq 2l
2a(1 + l2) + (1 - l2)
\bigl(
1 + | s| 2
\bigr) - k
2a
(4.6)
with the condition
2l
2a(1 + l2) + (1 - l2)
\bigl(
1 + | s| 2
\bigr) - k
2a
\geq 0. (4.7)
For the case, when k = 0, the condition (4.7) holds for every l, while (4.6) is satisfied for l1 \leq l < 1,
where
l1 =
| s - 1| 2
| \=s2 - 1|
.
If, on the other hand, s = 1 and k \in (0, 1), then (4.7) and (4.6) hold for k \leq l < 1. Assume now
s \not = 1 and k \in (0, 1). The condition (4.7) reduces to the quadratic inequality
l2
\bigl[
k(1 + | s| 2) - 2ak
\bigr]
+ 4al - k
\bigl[
2a+ 1 + | s| 2
\bigr]
\geq 0
or
kl2| s - 1| 2 + 4al - k| s+ 1| 2 \geq 0. (4.8)
Therefore, we find that (4.7) (or (4.8)) holds for l2 \leq l < 1, where
l2 =
\sqrt{}
4a2 + k2| \=s2 - 1| 2 - 2a
k| s - 1| 2
.
Similarly, (4.7) may be rewritten as
(1 - l2)| \=s2 - 1| \leq 4al - 2ak(1 + l2) - k(1 - l2)(1 + | s| 2)
or
l2
\bigl[
k| s - 1| 2 + | \=s2 - 1|
\bigr]
+ 4al - k| s+ 1| 2 - | \=s2 - 1| \geq 0,
that is, satisfied for l3 \leq l < 1, where
l3 =
| s - 1| 2 + k| \=s2 - 1|
| \=s2 - 1| + k| s - 1| 2
.
We note that l2 \leq l3 . Indeed, it is trivial that\bigl[
| \=s2 - 1| + k| s - 1| 2
\bigr] \sqrt{}
4a2 + k2| \=s2 - 1| 2 \leq
\bigl[
| \=s2 - 1| + k| s - 1| 2
\bigr] \bigl[
2a+ k| \=s2 - 1|
\bigr]
.
Moreover, we see at once that\bigl[
| \=s2 - 1| + k| s - 1| 2
\bigr] \bigl[
2a+ k| \=s2 - 1|
\bigr]
\leq
\bigl[
| \=s2 - 1| + k| s - 1| 2
\bigr] \bigl[
2a+ k| \=s2 - 1|
\bigr]
+ 4ak| s - 1| 2.
Combining the last two inequalities, we obtain\bigl[
| \=s2 - 1| + k| s - 1| 2
\bigr] \sqrt{}
4a2 + k2| \=s2 - 1| 2 \leq
\bigl[
| \=s2 - 1| + k| s - 1| 2
\bigr] \bigl[
2a+ k| \=s2 - 1|
\bigr]
+4ak| s - 1| 2,
which is equivalent to the desired inequality l2 \leq l3 . Likewise, it is a simple matter to show that
l3 < 1, and the proof is complete, by setting K := l3 . We note also, that the case k = 0 may be
included to the last case (i.e., s \not = 1).
Theorem 4.1 is proved.
Several similar sufficient conditions for quasiconformal extensions as in the Theorem 4.1 can be
derived. Here we select a few example out of a large variety of possibilities. The following is based
on the Theorem 2.2.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
34 E. DENIZ, S. KANAS, H. ORHAN
Theorem 4.2. Let \alpha > 0 and f, g \in \scrA . If
z1 - \alpha g(z)\alpha - 1f \prime (z) \in U(k)
for all z \in \scrU , then the function G\alpha (z) can be extended to a k-quasiconformal automorphism of \BbbC .
Proof. Set
\scrL (z, t) =
\left( \alpha z\int
0
g\alpha - 1(u)f \prime (u)du+ (e\alpha t - 1)z\alpha
\right) 1/\alpha
.
An easy computation shows
p(z, t) =
1
e\alpha t
\bigl(
z1 - \alpha g(z)\alpha - 1f \prime (z)
\bigr)
+
\biggl(
1 - 1
e\alpha t
\biggr)
,
and the assertion follows by the same methods as in Theorem 4.1, applying Theorems 2.1 and 2.2.
In the same manner, by definition of the suitable Loewner chain, several univalence criterion may
by found. For example, the condition
zG\prime
\alpha (z)
G\alpha (z)
\in U(k), \alpha \in \BbbC ,
which is based on the integral operator G\alpha (z), is given by the Loewner chain
\scrL (z, t) = etG\alpha (z).
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after revision — 06.04.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
|
| id | umjimathkievua-article-1148 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:53Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/8fe7936a2d4ab3e4f47b430f6a957294.pdf |
| spelling | umjimathkievua-article-11482022-03-27T15:39:11Z Univalence criteria and quasiconformal extension of a general integral operator Univalence criteria and quasiconformal extension of a general integral operator Deniz, E. Kanas, S. Orhan, H. Deniz, E. Kanas, S. Orhan, H. KANAS, STANIS LAWA ORHAN, HALIT Univalent function; Quasiconformal mapping; Univalence condition; Integral operator; Loewner chain. UDC 517.5 We give some sufficient conditions of analyticity and univalence for functions defined by an integral operator. Next, we refine the result to a quasiconformal extension criterion with the help of the Becker’s method. Further, new univalence criteria and the significant relationships with other results are given. A number of known univalence conditions would follow upon specializing the parameters involved in main results. &nbsp; УДК 517.5 Унiвалентнi критерiї та квазiконформне розширення iнтегрального оператора загального вигляду Запропоновано достатнi умови аналiтичностi та унiвалентностi для функцiй, що визначаються деяким iнтегральним оператором. Цей результат зводиться до критерiю квазiконформного розширення за допомогою методу Бекера. Далi отримано новi критерiї унiвалентностi та вказано важливi зв’язки з iншими результатами. Також з основного результату при рiзних значеннях параметрiв, якi задiянi у формулюваннi цього результату, випливають деякi вже вiдомi умови унiвалентностi. Institute of Mathematics, NAS of Ukraine 2022-01-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1148 10.37863/umzh.v74i1.1148 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 1 (2022); 24 - 35 Український математичний журнал; Том 74 № 1 (2022); 24 - 35 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1148/9173 Copyright (c) 2021 Erhan Deniz, STANIS LAWA KANAS, HALIT ORHAN |
| spellingShingle | Deniz, E. Kanas, S. Orhan, H. Deniz, E. Kanas, S. Orhan, H. KANAS, STANIS LAWA ORHAN, HALIT Univalence criteria and quasiconformal extension of a general integral operator |
| title | Univalence criteria and quasiconformal extension of a general integral operator |
| title_alt | Univalence criteria and quasiconformal extension of a general integral operator |
| title_full | Univalence criteria and quasiconformal extension of a general integral operator |
| title_fullStr | Univalence criteria and quasiconformal extension of a general integral operator |
| title_full_unstemmed | Univalence criteria and quasiconformal extension of a general integral operator |
| title_short | Univalence criteria and quasiconformal extension of a general integral operator |
| title_sort | univalence criteria and quasiconformal extension of a general integral operator |
| topic_facet | Univalent function Quasiconformal mapping Univalence condition Integral operator Loewner chain. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1148 |
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