Volterra-type operator on the subclasses of univalent functions
UDC 517.5 In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order $b$ $(b\neq 0)$ and spirallike functions of type $ \lambda$ $\Big( {-\dfrac{\pi}{2}}<\lambda<\dfrac{\pi}{2} \Big)$...
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In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order $b$ $(b\neq 0)$ and spirallike functions of type $ \lambda$ $\Big( {-\dfrac{\pi}{2}}<\lambda<\dfrac{\pi}{2} \Big)$ with the complex order $b$ $(b\neq 0).$ We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes.
In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk $ \mathbb{D}=\lbrace z\in \mathbb{C}:\vert z\vert<1\rbrace $ under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces. |
| doi_str_mv | 10.37863/umzh.v74i1.1116 |
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DOI: 10.37863/umzh.v74i1.1116
UDC 517.5
M. Mahboobi (Payme Noor Univ., Tehran, Iran)
VOLTERRA-TYPE OPERATOR
ON THE SUBCLASSES OF UNIVALENT FUNCTIONS
ОПЕРАТОРИ ТИПУ ВОЛЬТЕРРА
НА ПIДКЛАСАХ УНIВАЛЕНТНИХ ФУНКЦIЙ
In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and
convex functions of complex order b (b \not = 0) and spirallike functions of type \lambda
\Bigl(
- \pi
2
< \lambda <
\pi
2
\Bigr)
with the complex order
b (b \not = 0). We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to
the mentioned classes.
In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the
open unit disk \BbbD = \{ z \in \BbbC : | z| < 1\} under the action of a Volterra-type operator and the product of the composition
operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of
the Schwartzian norm of the above operators in these spaces.
Вивчаються необхiднi та достатнi умови належностi елемента до класу зiркоподiбних та опуклих функцiй комп-
лексного порядку b (b \not = 0) i спiралеподiбних функцiй типу \lambda
\Bigl(
- \pi
2
< \lambda <
\pi
2
\Bigr)
комплексного порядку b (b \not = 0).
Встановлено точнi оцiнки для коефiцiєнта другого члена ряду Тейлора для функцiй iз вказаних класiв.
У основнiй частинi роботи отримано необхiднi та достатнi умови обмеженостi образу вiдкритого одиничного
диска \BbbD = \{ z \in \BbbC : | z| < 1\} пiд дiєю оператора типу Вольтерра i добутку оператора композицiї та оператора
типу Вольтерра у просторi унiвалентних функцiй та його пiдпросторi. Насамкiнець встановлено оцiнки для норми
Шварца згаданих операторiв у цих просторах.
1. Introduction. The convolution or Hadamard product of two power series functions f(z) =
=
\sum \infty
n=0
anz
n and g(z) =
\sum \infty
n=0
bnz
n is defined as the power series (f \ast g) =
\sum \infty
n=0
anbnz
n.
Let \scrH (\BbbD ) denote the class of all functions holomorphic in the unit disk \BbbD , \BbbD = \{ z \in \BbbC :
| z| < 1\} , of the complex plan \BbbC . Let \scrA be the class of functions f(z) of the form
f(z) = z +
\infty \sum
n=2
anz
n,
which are analytic in the open unit disk \BbbD . Thus the class \scrA is a subclass of \scrH (\BbbD ).
Furthermore, let \scrP denote the class of functions p(z) of the form
p(z) = 1 +
\infty \sum
n=1
pnz
n,
which are analytic in \BbbD . If p(z) \in \scrP satisfies \Re (p(z)) > 0, z \in \BbbD , then we say that p(z) is the
Caratheodory function (see [4]).
If f(z) \in \scrA satisfies the inequality
\Re
\biggl(
zf \prime (z)
f(z)
\biggr)
> \alpha , z \in \BbbD ,
for some \alpha , 0 \leq \alpha < 1, then f(z) is said to be starlike of order \alpha in \BbbD . We denote by \scrS \ast (\alpha ) the
subclass of \scrA consisting of functions f(z), which are starlike of order \alpha in \BbbD . Similarly, we say
c\bigcirc M. MAHBOOBI, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1 77
78 M. MAHBOOBI
that f(z) is a member of the class \scrK (\alpha ) of convex functions of order \alpha in \BbbD if f(z) \in \scrA satisfies
the following inequality:
\Re
\biggl(
1 +
zf \prime \prime (z)
f \prime (z)
\biggr)
> \alpha , z \in \BbbD ,
for some \alpha , 0 \leq \alpha < 1.
As usual, in the present investigation, we write
\scrS \ast = \scrS \ast (0) and \scrK = \scrK (0).
Moreover, for some non-zero complex numbers b, we consider the subclasses \scrS \ast
b and \scrK b of \scrA
as follows:
\scrS \ast
b =
\biggl\{
f(z) \in \scrA : \Re
\biggl[
1 +
1
b
\biggl(
zf \prime (z)
f(z)
- 1
\biggr) \biggr]
> 0, z \in \BbbD
\biggr\}
and
\scrK b =
\biggl\{
f(z) \in \scrA : \Re
\biggl[
1 +
1
b
\biggl(
zf \prime \prime (z)
f \prime (z)
\biggr) \biggr]
> 0, z \in \BbbD
\biggr\}
.
Then we can see that
\scrS \ast
1 - \alpha = \scrS \ast (\alpha ) and \scrK 1 - \alpha = \scrK (\alpha ).
Let f be a function analytic and locally univalent in the unit disk \BbbD ,
Sf = (f \prime \prime /f \prime )\prime - 1/2(f \prime \prime /f \prime )2
denote its Schwarzian derivative and
\| Sf (z)\| = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
(1 - | z| 2)2| Sf |
denote its Schwarzian norm.
Recall first that if f maps the disk conformally onto a convex region, then the function
g(z) = 1 +
zf \prime \prime (z)
f \prime (z)
has positive real part in \BbbD (see, for instance, [3]). Since g(0) = 1, this say that g is subordinate to
the half-plan mapping \scrL (z) = (1+ z)/(1 - z), so that g(z) = \scrL (\varphi (z)) for some Schwarz functions
\varphi . In other words,
zf \prime \prime (z)
f \prime (z)
=
1 + \varphi (z)
1 - \varphi (z)
- 1 =
2\varphi (z)
1 - \varphi (z)
,
where \varphi is analytic and has the property | \varphi (z)| \leq | z| in \BbbD . With the notation \psi (z) =
\varphi (z)
z
this
gives the representation
f \prime \prime (z)
f \prime (z)
=
2\psi (z)
1 - z\psi (z)
(1)
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
VOLTERRA-TYPE OPERATOR ON THE SUBCLASSES OF UNIVALENT FUNCTIONS 79
for the pre-Schwarzian, where \psi is analytic and satisfies | \psi (z)| \leq 1 in \BbbD . Straight forward calcula-
tion now gives the Schwarzian of f in the form
Sf (z) =
\biggl(
f \prime \prime (z)
f \prime (z)
\biggr) \prime
- 1
2
\biggl(
f \prime \prime (z)
f \prime (z)
\biggr) 2
=
2\psi \prime (z)
(1 - z\psi (z))2
.
But | \psi \prime (z)| \leq
\bigl(
1 - | \psi (z)| 2
\bigr)
/(1 - | z| 2) by the invariant form of the Schwarz lemma, so we conclude
that
| Sf (z)| \leq 2
1 - | \psi (z)| 2
(1 - | z| 2) (1 - | z\psi (z)| )2
\leq 2
(1 - | z| 2)2
. (2)
In other words, the inequality (2) says that the Schwarzian norm
\| Sf (z)\| = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
(1 - | z| 2)2| Sf | ,
of convex mapping is no large than 2. The bound is best possible since the parallel strip mapping
L(z) =
1
2
\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
1 + z
1 - z
\biggr)
,
has Schwarzian SL(z) = 2(1 - z2) - 2. Nehari [12] also stated that \| Sf (z)\| < 2 if the convex
mapping f is bounded.
The composition operators C\varphi are defined on \scrH (\BbbD ) as follows:
C\varphi (f) = f \circ \varphi , f \in \scrH (\BbbD ),
for some self-maps \varphi : \BbbD \rightarrow \BbbD of the unit disk \BbbD .
For g \in \scrH (\BbbD ), the integral operator Ig
Igh(z) =
z\int
0
h\prime (\xi )g(\xi )d\xi , h \in \scrH (\BbbD ),
was introduced in [14] and is called the Volterra-type operator.
In this paper, we introduced some new subclasses of \scrH (\BbbD ) as follows:
P (\beta , b) :=
\biggl\{
p(z) \in \scrP : \Re
\biggl[
1
b
\biggl(
zp\prime (z)
p(z)
\biggr) \biggr]
\geq \beta
\biggr\}
and
P \prime (\beta , b) :=
\biggl\{
p(z) \in \scrP : \Re
\biggl[
1
b
\biggl(
zp\prime (z)
p(z)
\biggr) \biggr]
\leq \beta
\biggr\}
for some real numbers \beta and non-zero complex numbers b.
Example. Let
p(z) =
1
1 - z
= 1 + z + z2 + . . . \in P ( - 1/2, 1),
p(z) =
1
1 + z
= 1 - z + z2 - z3 \pm . . . \in P ( - 1/2, 1),
p(z) = 1 + z \in P \prime (1/2, 1), p(z) = 1 - z \in P \prime (1/2, 1).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
80 M. MAHBOOBI
Moreover, for some non-zero complex numbers b and real \lambda , - \pi
2
< \lambda <
\pi
2
, we define the subclass
\^\scrS \lambda (b) of \scrA as follows:
\^\scrS \lambda (b) :=
\biggl\{
f(z) \in \scrA : \Re
\biggl[
ei\lambda
\biggl(
1 +
1
b
\biggl(
zf \prime (z)
f(z)
- 1
\biggr) \biggr) \biggr]
> 0, z \in \BbbD
\biggr\}
.
If a function f(z) belong to the class \^\scrS \lambda (b), we say that f(z) is spirallike of type \lambda with the
complex order b, b \not = 0.
In this paper, we get some properties for the functions in \scrS \ast
b , \scrK b and \^\scrS \lambda (b). Also we study the
Volterra-type operator Ig on \scrK and \scrK b. Furthermore, we get necessary and sufficient condition such
that Igh(\BbbD ) is bounded, moreover, we obtain the sufficient condition such that \| Sf (z)\| < 2.
In addition to the references referred to in this article, to better understand the properties of the
spaces and operators mentioned in this work, we can study references [1, 2, 6, 13].
2. Some of the new properties of the functions belong to the subclasses of starlike and
convex functions. In this section, we examine some of the properties of the functions belong to
the subclasses of starlike and convex functions of complex order b, b \not = 0, and spirallike functions
of type \lambda , - \pi
2
< \lambda <
\pi
2
, with the complex order b, b \not = 0. Among the items of interest are the
necessary and sufficient conditions for a member to belong to the classes \scrS \ast
b , \scrK b and \^\scrS \lambda (b). In the
following, we obtain the sharp estimates for the second coefficients of the Taylor series of functions
belonging to the mentioned classes.
Theorem 2.1. Let b \in \BbbC , b \not = 0, \lambda \in
\Bigl(
- \pi
2
,
\pi
2
\Bigr)
and \beta = e - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda . Then f belongs to \^\scrS \lambda (b)
if and only if there is g \in \scrS \ast
b such that
f(z) = z
\biggl(
g(z)
z
\biggr) \beta
. (3)
The branch of the power function is choosen such that
\biggl(
g(z)
z
\biggr) \beta \bigm| \bigm| \bigm| \bigm|
z=0
= 1.
Proof. First assume f \in \^\scrS \lambda (b). Cleary the relation (3) is equivalent to
g(z) = z
\biggl(
f(z)
z
\biggr) ei\lambda
cos\lambda
, z \in \BbbD ,
we choose the branch of the power function such that
\biggl(
f(z)
z
\biggr) ei\lambda
cos\lambda
\bigm| \bigm| \bigm| \bigm|
z=0
= 1. A simple computation
yields the relation
1 +
1
b
\biggl(
zg\prime (z)
g(z)
- 1
\biggr)
= (1 + i \mathrm{t}\mathrm{a}\mathrm{n}\lambda )
\biggl(
1 +
zf \prime (z)
bf(z)
\biggr)
- 1 + i \mathrm{t}\mathrm{a}\mathrm{n}\lambda
b
- i \mathrm{t}\mathrm{a}\mathrm{n}\lambda .
Therefore,
\Re
\biggl[
1 +
1
b
\biggl(
zg\prime (z)
g(z)
- 1
\biggr) \biggr]
=
1
\mathrm{c}\mathrm{o}\mathrm{s}\lambda
\Re
\biggl[
ei\lambda
\biggl(
1 +
1
b
\biggl(
zg\prime (z)
g(z)
- 1
\biggr) \biggr) \biggr]
.
Since f \in \^\scrS \alpha (b), consequently g is starlike of complex order b.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
VOLTERRA-TYPE OPERATOR ON THE SUBCLASSES OF UNIVALENT FUNCTIONS 81
Conversely, if g \in \scrS \ast
b , then in view of the above relation and the fact that \lambda \in
\Bigl(
- \pi
2
,
\pi
2
\Bigr)
, one
deduces that
\Re
\biggl[
ei\lambda
\biggl(
1 +
1
b
\biggl(
zg\prime (z)
g(z)
- 1
\biggr) \biggr) \biggr]
> 0, z \in \BbbD .
Thus, f is spirallike of type \lambda with complex order b.
Theorem 2.1 is proved.
Theorem 2.2. Let b \in \BbbC and b \not = 0. Then we have the equality
\scrS \ast
b =
\bigl\{
zh\prime (z) : h \in \scrK b
\bigr\}
.
Proof. Let f(z) = z +
\sum \infty
n=2
anz
n \in \scrS \ast
b . It’s obvious that
f(z) = z
\Biggl(
1 +
\infty \sum
n=2
anz
n - 1
\Biggr)
.
We put
h\prime (z) = 1 +
\infty \sum
n=2
anz
n - 1,
therefore,
h(z) = z +
\infty \sum
n=2
1
n
anz
n.
Then f(z) = zh\prime (z). Applying that h(z) \in \scrK b if and only if zh\prime (z) \in \scrS \ast
b , we deduced that h(z)
belongs to \scrK b.
Theorem 2.2 is proved.
Theorem 2.3. For the function f(z) \in \scrA , it follows that
f(z) \in \scrS \ast
b \Leftarrow \Rightarrow z
\biggl(
f(z)
z
\biggr) 1
b
\in \scrS \ast , b \in \BbbC , b \not = 0.
Proof. Let f(z) be a starlike of complex order b. By using Theorem 2.2, there is h \in \scrK b such
that f(z) = zh\prime (z). Since h \in \scrK b, then, by using Theorem 1.2 in [5], we have z (h\prime (z))
1
b \in \scrS \ast .\biggl(
Another proof for this theorem. We set F (z) = z
\biggl(
f(z)
z
\biggr) 1
b
. Therefore,
\Re
\biggl[
zF \prime (z)
F (z)
\biggr]
= \Re
\biggl[
1 +
1
b
\biggl(
zf \prime (z)
f(z)
- 1
\biggr) \biggr]
> 0,
and then f \in \scrS \ast
b .
\biggr)
By using Theorems 2.1 and 2.2, we have the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
82 M. MAHBOOBI
Theorem 2.4. Let b \in \BbbC and b \not = 0, \lambda \in
\Bigl(
- \pi
2
,
\pi
2
\Bigr)
and \beta = ei\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda . Then f belongs to
\^\scrS \lambda (b) if and only if there is h \in \scrK b such that
f(z) = z
\bigl(
h\prime (z)
\bigr)
.
Theorem 2.5. Let f(z) = z + a2z
2 + a3z
3 + . . . be a starlike function of complex order b,
b \not = 0. Then | a2| \leq 2| b| . This bound is sharp. Equality is attained for fb(z) =
z
(1 - z)2b
.
Proof. Let f(z) belongs to \scrS \ast
b . By using Theorem 2.3, we have g(z) = z
\biggl(
f(z)
z
\biggr) 1
b
\in \scrS \ast . Let
g(z) = z + b2z
2 + b3z
3 + . . . , therefore b2 =
1
b
a2. So, by using Bieberbach theorem, we have
| a2| = | b| | b2| \leq 2| b| . Since
z
(1 - z)2b
= z +
\infty \sum
n=2
\prod n
j=1
(j + 2(b - 1))
(n - 1)!
zn,
then it is obvious that equality is attained for fb.
Theorem 2.5 is proved.
Theorem 2.6. Let f(z) = z+a2z
2+a3z
3+. . . be a spirallike function of type \lambda , \lambda \in
\Bigl(
- \pi
2
,
\pi
2
\Bigr)
,
with complex order b, b \not = 0. Then | a2| \leq 2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Proof. Let f(z) \in \^\scrS \lambda (b). By Theorem 2.1, there is g(z) = z+ b2z
2 + b3z
3 + . . . \in \scrS \ast
b such that
f(z) = z
\biggl(
g(z)
z
\biggr) \beta
, \beta = e - i\lambda \mathrm{c}\mathrm{o}\mathrm{s}\lambda . Then a2 = b2\beta , so | a2| = | b2| \mathrm{c}\mathrm{o}\mathrm{s}\lambda . By using Theorem 2.5,
we have | a2| \leq 2| b| \mathrm{c}\mathrm{o}\mathrm{s}\lambda .
Remark. Since \^\scrS \lambda (1) = \^\scrS \lambda , we obtain Corollary 2.4.12 in [7] as a result of the above theorem.
3. Volterra-type operator on subclasses of convex functions. Here, first, we express and prove
two lemmas that are widely used in proving the main theorems of this paper.
Lemma 3.1. 1. Let b \in \BbbC , b \not = 0 and \beta \geq 0. If g belongs to P (\beta , b), then Ig is an operator
on \scrK b.
2. Let \beta \in \BbbR , 0 \leq \alpha < 1 and 0 \leq \alpha + \beta < 1. If g belongs to P (\beta , 1), then Ig is an operator
from \scrK (\alpha ) to \scrK (\alpha + \beta ).
Proof. Due to the similarity of the proof of parts 1 and 2, so we only do the proof of part 1.
Let h \in \scrK b, then
\Re
\biggl[
1 +
1
b
\biggl(
z (Igh)
\prime \prime (z)
(Igh)
\prime (z)
\biggr) \biggr]
= \Re
\biggl[
1 +
1
b
\biggl(
zh\prime \prime (z)g(z) + zh\prime (z)g\prime (z)
h\prime (z)g(z)
\biggr) \biggr]
=
= \Re
\biggl[
1 +
1
b
\biggl(
zh\prime \prime (z)
h\prime (z)
+
zg\prime (z)
g(z)
\biggr) \biggr]
=
= \Re
\biggl[
1 +
1
b
\biggl(
zh\prime \prime (z)
h\prime (z)
\biggr) \biggr]
+ \Re
\biggl[
1
b
\biggl(
zg\prime (z)
g(z)
\biggr) \biggr]
. (4)
By hypothesis of this lemma and relation (4), we have
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
VOLTERRA-TYPE OPERATOR ON THE SUBCLASSES OF UNIVALENT FUNCTIONS 83
\Re
\biggl[
1 +
1
b
\biggl(
z (Igh)
\prime \prime (z)
(Igh)
\prime (z)
\biggr) \biggr]
> 0,
therefore, Igh belongs to \scrK b for each h \in \scrK b.
Lemma 3.2. The function f is convex of complex order b, b \not = 0 in \BbbD if and only if
f \prime \ast
z
\biggl(
1 + x
2b
\biggr)
+ 1 - z
(1 - z)2
\not = 0, z \in \BbbD , | x| = 1.
Proof. The function f is convex of complex order b if and only if
\Re
\biggl[
1 +
1
b
\biggl(
zf \prime \prime (z)
f \prime (z)
\biggr) \biggr]
> 0, z \in \BbbD . (5)
By Lemma 1 in [8], relation (5) is equivalent to
1 +
1
b
\biggl(
(zf \prime (z))\prime
f \prime (z)
- 1
\biggr)
\not = x - 1
x+ 1
, z \in \BbbD , | x| = 1, x \not = - 1,
which simplifies to
(1 + x)
\bigl(
zf \prime (z)
\bigr) \prime
+ (2b - x - 1)f \prime (z) \not = 0.
We have
(1 + x)
\bigl(
zf \prime (z)
\bigr) \prime
= f \prime (z) \ast 1 + x
(1 - z)2
and
(2b - x - 1)f \prime (z) = f \prime (z) \ast 2b - x - 1
1 - z
,
so that
(1 + x)
\bigl(
zf \prime (z)
\bigr) \prime
+ (2b - x - 1)f \prime (z) = f \prime (z) \ast 1 + x
(1 - z)2
+ f \prime (z) \ast 2b - x - 1
1 - z
=
= f \prime (z) \ast (1 + x) + (1 - z)(2b - x - 1)
(1 - z)2
=
= f \prime (z) \ast (1 + x - 2b)z + 2b
(1 - z)2
\not = 0.
Since b \not = 0, we get
f \prime \ast
z
\biggl(
1 + x - 2b
2b
\biggr)
+ 1
(1 - z)2
\not = 0, z \in \BbbD , | x| = 1, x \not = - 1.
The case x = 1 in the convolution condition is equivalent to stating f \prime \not = 0 for each z \in \BbbD , which
is a necessary condition for univalence.
We now obtain the necessary and sufficient conditions to boundedness the image of the open unit
disk \BbbD = \{ z \in \BbbC : | z| < 1\} under the effect of the Volterra-type operator on the space consisting of
univalent functions and its subspace, and finally we get an estimate of the Schwartzian norm for the
above-mentioned operator on class \scrA and subclass \scrK (\alpha ) of it, where 0 \leq \alpha < 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
84 M. MAHBOOBI
Theorem 3.1. Let \beta \in \BbbR , 0 \leq \alpha < 1, \alpha + \beta \geq 0 and g \in P (\beta , 1) and h \in \scrK (\alpha ). Then the
image (Igh)(\BbbD ) is bounded if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| 2 + zh\prime \prime (z)
h\prime (z)
+
zg\prime (z)
g(z)
\bigm| \bigm| \bigm| \bigm| < 1. (6)
Proof. By using Lemma 3.1 (part 2), we have that Igh belongs to \scrK . In relation (1), we put
f = Igh. Then there is an analytic function \psi such that
(Igh)
\prime \prime
Igh)\prime
=
2\psi (z)
1 - z\psi (z)
,
therefore
\psi (z) =
(Igh)
\prime \prime (z)/(Igh)
\prime (z)
2 + z(Igh)\prime \prime (z)/(Igh)\prime (z)
=
g\prime (z)/g(z) + h\prime \prime (z)/h\prime (z)
2 + zg\prime (z)/g(z) + zh\prime \prime (z)/h\prime (z)
.
We have
1 - | z|
| 1 - z\psi (z)|
=
1 - | z| \bigm| \bigm| \bigm| \bigm| 1 - zg\prime (z)/g(z) + zh\prime \prime (z)/h\prime (z)
2 + zg\prime (z)/g(z) + zh\prime \prime (z)/h\prime (z)
\bigm| \bigm| \bigm| \bigm| =
=
1 - | z|
2
2 + zg\prime (z)/g(z) + zh\prime \prime (z)/h\prime (z)
=
=
1
2
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| 2 + zh\prime \prime (z)
h\prime (z)
+
zg\prime (z)
g(z)
\bigm| \bigm| \bigm| \bigm| .
By Theorem 2 in [3], we get that the image (Igh)(\BbbD ) is bounded if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
1 - | z|
| 1 - z\psi (z)|
<
1
2
.
By relation (6), the proof is complete.
Now we want to state a theorem that is similar to the previous theorem in proof. The main
difference is the use of Lemma 3.2 instead of Lemma 3.1 in proof.
Theorem 3.2. Let g \in \scrP and h \in \scrA such that (h\prime g) (z) \ast (x - 1)z
2(1 - z)2
\not = 0. Then the image
(Igh)(\BbbD ) is bounded if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| 2 + zh\prime \prime (z)
h\prime (z)
+
zg\prime (z)
g(z)
\bigm| \bigm| \bigm| \bigm| < 1.
By using Lemma 3.1 (part 2), we have the following corollary.
Corollary 3.1. Let \beta \in \BbbR , 0 \leq \alpha < 1, \alpha + \beta \geq 0. If g \in P\beta and h \in \scrK (\alpha ), then
\| SIgh\| \leq 2.
By using Lemma 3.2, we obtain the following corollary.
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VOLTERRA-TYPE OPERATOR ON THE SUBCLASSES OF UNIVALENT FUNCTIONS 85
Corollary 3.2. Let g \in \scrP and h \in \scrA . If (h\prime g) (z) \ast (x - 1)z
2(1 - z)2
\not = 0, then
\| SIgh\| \leq 2.
By using Theorem 3.1, we get the following corollary.
Corollary 3.3. Let \beta \in \BbbR , 0 \leq \alpha < 1, \alpha + \beta \geq 0 and g \in P\beta and h \in \scrK (\alpha ). If
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| 2 + zh\prime \prime (z)
h\prime (z)
+
zg\prime (z)
g(z)
\bigm| \bigm| \bigm| \bigm| < 1,
then
\| SIgh\| < 2.
By using Theorem 3.2, we have the following corollary.
Corollary 3.4. Let g \in \scrP and h \in \scrA such that (h\prime g) (z) \ast (x - 1)z
2(1 - z)2
\not = 0. If
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| 2 + zh\prime \prime (z)
h\prime (z)
+
zg\prime (z)
g(z)
\bigm| \bigm| \bigm| \bigm| < 1,
then
\| SIgh\| < 2.
4. Product of composition operators and Volterra-type operator on subclasses of convex
functions. Products of composition operators and integral-type operators have been recently in-
troduced by S. Li and S. Stević in [9 – 11]. Here, we shall be interested in studing the product of
composition operators and Volterra-type integral operators, which are defined by
(C\sigma (Igh)) (z) =
\sigma (z)\int
0
h\prime (\xi )g(\xi )d\xi , z \in \BbbD ,
on subclasses of \scrA , where g \in \scrA and \sigma is an analytic self-map of the unit disk. In this section, we
assume that \sigma (z) be the Mobius automorphism \sigma (z) =
z + z0
1 + \=z0z
on \BbbD , where, z0 be the fixed point
in \BbbD .
In fact, in the final section of this paper, we intend to examine a similar discussion of the previous
section for the said operator.
Lemma 4.1. Let \beta \in \BbbR , 0 \leq \alpha < 1 and 0 \leq \alpha + \beta < 1. If g belongs to P (\beta , 1), then C\sigma Ig is
an operator from \scrK (\alpha ) to \scrK .
Proof. By hypothesis of this lemma and by Lemma 3.1 (part 2), it is obvious that Ig is an
operator from \scrK (\alpha ) to \scrK . Therefore Igh is a convex map. Let f = Igh. By Lemma 1 in [7], we
have fo\sigma is a convex mapping of \BbbD . We know that
fo\sigma (z) = f (\sigma (z)) = (Igh) (\sigma (z)) =
\sigma (z)\int
0
h\prime (\xi )g(\xi )d\xi = (C\sigma Igh)(z).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
86 M. MAHBOOBI
Lemma 4.2. Let g \in \scrP and h \in \scrA . If\bigl(
h\prime g
\bigr)
(z) \ast (x - 1)z
2(1 - z)2
\not = 0,
then C\sigma Igh \in \scrK .
By using Lemma 3.2 and Lemma 1 in [3], we have the following theorem.
Theorem 4.1. Let \beta \in \BbbR , 0 \leq \alpha < 1, \alpha + \beta \geq 0 and g \in P (\beta , 1) and h \in \scrK (\alpha ). Then the
image (C\sigma Igh)(\BbbD ) is bounded if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| (\sigma (z) - z0)A(z) + 2
(\sigma (z) - z0 + \=z0z\sigma (z) - z)A(z) + 2\=z0z + 2
\bigm| \bigm| \bigm| \bigm| < 1
2
, (7)
where A(z) =
g\prime (\sigma (z))
g (\sigma (z))
+
h\prime \prime (\sigma (z))
h\prime (\sigma (z))
.
Proof. By using Lemma 4.1, we have (C\sigma Igh) \in \scrK . In the proof of Theorem 3.1 we saw that
\psi (z) =
g\prime (z)/g(z) + h\prime \prime (z)/h\prime (z)
2 + zg\prime (z)/g(z) + zh\prime \prime (z)/h\prime (z)
.
By using Lemma 1 in [3], there is an analytic function \lambda such that
(C\sigma Igh)
\prime \prime
(C\sigma Igh)\prime
=
2\lambda (z)
1 - z\lambda (z)
,
where
\lambda (z) =
\psi (\sigma (z)) - \=z0
1 - z0\psi (\sigma (z))
.
We have
z\lambda (z) =
zA(z)
2 + \sigma (z)A(z)
- \=z0z
1 - z0A(z)
2 + \sigma (z)A(z)
=
(z - \=z0z\sigma (z))A(z) - 2\=z0z
(\sigma (z) - z0)A(z) + 2
,
therefore,
1 - | z|
| 1 - z\lambda (z)|
=
1 - | z| \bigm| \bigm| \bigm| \bigm| 1 - (z - \=z0z\sigma (z))A(z) - 2\=z0z
(\sigma (z) - z0)A(z) + 2
\bigm| \bigm| \bigm| \bigm| =
= (1 - | z| )
\bigm| \bigm| \bigm| \bigm| (\sigma (z) - z0)A(z) + 2
(\sigma (z) - z0 + \=z0z\sigma (z) - z)A(z) + 2\=z0z + 2
\bigm| \bigm| \bigm| \bigm| .
By using Theorem 2 in [3], we get the image (C\sigma Igh)(\BbbD ) is bounded if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
1 - | z|
| 1 - z\lambda (z)|
<
1
2
.
By relation (7), the proof is complete.
In the following, we state a theorem that has a similar proof to the previous theorem. The only
major difference is the use of Lemma 4.2 instead of Lemma 4.1 during the proof.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
VOLTERRA-TYPE OPERATOR ON THE SUBCLASSES OF UNIVALENT FUNCTIONS 87
Theorem 4.2. Let g \in \scrP and h \in \scrA such that (h\prime g) (z) \ast (x - 1)z
2(1 - z)2
\not = 0. Then the image
(C\sigma Igh)(\BbbD ) is bounded if and only if
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| (\sigma (z) - z0)A(z) + 2
(\sigma (z) - z0 + \=z0z\sigma (z) - z)A(z) + 2\=z0z + 2
\bigm| \bigm| \bigm| \bigm| < 1
2
,
where A(z) =
g\prime (\sigma (z))
g (\sigma (z))
+
h\prime \prime (\sigma (z))
h\prime (\sigma (z))
.
By using Lemma 4.1, we have the following corollary.
Corollary 4.1. Let \beta \in \BbbR , 0 \leq \alpha < 1, \alpha + \beta \geq 0. If g \in P (\beta , 1) and h \in \scrK (\alpha ), then
\| SC\sigma Igh\| \leq 2.
By using of the Lemma 4.2, we obtain the following corollary.
Corollary 4.2. Let g \in \scrP and h \in \scrA . If
\bigl(
h\prime g
\bigr)
(z) \ast (x - 1)z
2(1 - z)2
\not = 0,
then
\| SC\sigma Igh\| \leq 2.
By using Theorem 4.1, we get the following corollary.
Corollary 4.3. Let \beta \in \BbbR , 0 \leq \alpha < 1, \alpha + \beta \geq 0 and g \in P (\beta , 1) and h \in \scrK (\alpha ). If
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| (\sigma (z) - z0)A(z) + 2
(\sigma (z) - z0 + \=z0z\sigma (z) - z)A(z) + 2\=z0z + 2
\bigm| \bigm| \bigm| \bigm| < 1
2
,
where A(z) =
g\prime (\sigma (z))
g (\sigma (z))
+
h\prime \prime (\sigma (z))
h\prime (\sigma (z))
, then
\| SC\sigma Igh\| < 2.
By using Theorem 4.2, we have the following corollary.
Corollary 4.4. Let g \in \scrP and h \in \scrA such that (h\prime g) (z) \ast (x - 1)z
2(1 - z)2
\not = 0. If
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
| z| - \rightarrow 1
(1 - | z| )
\bigm| \bigm| \bigm| \bigm| (\sigma (z) - z0)A(z) + 2
(\sigma (z) - z0 + \=z0z\sigma (z) - z)A(z) + 2\=z0z + 2
\bigm| \bigm| \bigm| \bigm| < 1
2
,
where A(z) =
g\prime (\sigma (z))
g (\sigma (z))
+
h\prime \prime (\sigma (z))
h\prime (\sigma (z))
, then
\| SC\sigma Igh\| < 2.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
88 M. MAHBOOBI
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Received 25.02.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
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| id | umjimathkievua-article-1116 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:47Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1f/4ed3218ef529c57c708e792d9f79f81f.pdf |
| spelling | umjimathkievua-article-11162022-03-27T15:39:11Z Volterra-type operator on the subclasses of univalent functions Volterra-type operator on the subclasses of univalent functions Mahboobi, M. Mahboobi, M. operators convex functions of complex order starlike functions of complex order spirallike functions of type λ with complex order Schwazian derivative Schwazian norm composition operator Volterra type operator UDC 517.5 In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order $b$ $(b\neq 0)$ and spirallike functions of type $ \lambda$ $\Big( {-\dfrac{\pi}{2}}&lt;\lambda&lt;\dfrac{\pi}{2} \Big)$ with the complex order $b$ $(b\neq 0).$ We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes. In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk $ \mathbb{D}=\lbrace z\in \mathbb{C}:\vert z\vert&lt;1\rbrace $ under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces. UDC 517.5 Оператори типу Вольтерра на підкласах унівалентних функцій Вивчаються необхідні та достатні умови належності елемента до класу зіркоподібних та опуклих функцій комплексного порядку $b$ $(b\neq 0)$ і спіралеподібних функцій типу $\lambda$ $\Big( {-\dfrac{\pi}{2}}&lt;\lambda&lt;\dfrac{\pi}{2} \Big)$ комплексного порядку $ b$ $(b\neq 0).$ Встановлено точні оцінки для коефіцієнта другого члена ряду Тейлора для функцій із вказаних класів. У основній частині роботи отримано необхідні та достатні умови обмеженості образу відкритого одиничного диска $\mathbb{D}=\lbrace z\in \mathbb{C}:\vert z\vert&lt;1\rbrace $ під дією оператора типу Вольтерра і добутку оператора композиції та оператора типу Вольтерра у просторі унівалентних функцій та його підпросторі. Насамкінець встановлено оцінки для норми Шварца згаданих операторів у цих просторах. Institute of Mathematics, NAS of Ukraine 2022-01-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1116 10.37863/umzh.v74i1.1116 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 1 (2022); 77 - 88 Український математичний журнал; Том 74 № 1 (2022); 77 - 88 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1116/9181 Copyright (c) 2022 mahdi mahboobi |
| spellingShingle | Mahboobi, M. Mahboobi, M. Volterra-type operator on the subclasses of univalent functions |
| title | Volterra-type operator on the subclasses of univalent functions |
| title_alt | Volterra-type operator on the subclasses of univalent functions |
| title_full | Volterra-type operator on the subclasses of univalent functions |
| title_fullStr | Volterra-type operator on the subclasses of univalent functions |
| title_full_unstemmed | Volterra-type operator on the subclasses of univalent functions |
| title_short | Volterra-type operator on the subclasses of univalent functions |
| title_sort | volterra-type operator on the subclasses of univalent functions |
| topic_facet | operators convex functions of complex order starlike functions of complex order spirallike functions of type λ with complex order Schwazian derivative Schwazian norm composition operator Volterra type operator |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1116 |
| work_keys_str_mv | AT mahboobim volterratypeoperatoronthesubclassesofunivalentfunctions AT mahboobim volterratypeoperatoronthesubclassesofunivalentfunctions |