Certain Subordination and Superordination Properties of Analytic
Functions
MUSTAFA I. HAMEED1, ISSA ALKHARUSI2, ISRAA A. IBRAHIM3, WAFAA M. TAHA4, ALI
F. JAMEEL2, SUNDAS NAWAZ5, MOHAMMAD A. TASHTOUSH2,6,*
1Department of Mathematics, College of Education for Pure Sciences,
University of Anbar, Ramadi,
IRAQ
2Faculty of Education and Arts,
Sohar University, Sohar,
OMAN
3Ministry of Education, College of open Education,
Kirkuk Education Directorate, Kirkuk,
IRAQ
4Department of Mathematics, College of Sciences,
Kirkuk University, Kirkuk,
IRAQ
5Department of Mathematical Sciences,
Fatima Jinnah Women University,
Rawalpindi,
PAKISTAN
6Department of Basic Sciences, AL-Huson University College,
AL-Balqa Applied University,
AL-Salt,
JORDAN
*Corresponding Author
Abstract: - In this paper, we extract some subordination and Superordination properties using the characteristics
of the generalized byproduct operator. The article aims to demonstrate some applications of the differential
subordination concept to univalent function subclasses that contain specific convolutions as operators. During
this time, several highly complex mathematical detectives have emerged, including Riemann, Cauchy, Gauss,
Euler, and several others. Geometric function theory combines or involves geometry and analysis. The main
objectives of the paper above are to investigate the dependence principle and to introduce an extra subset over
polyvalent functions through a further operator related to higher-order derivative products. The results were
important when taking into account the numerous geometric characteristics, including radii over stiffness,
close-to-convexity, and convexity; value estimation; deformation and expansion bounds; and so on.
Key-Words: - Derivative Operator, Convex function, Subordination, Superordination, Univalent Function,
Analytic Functions.
Received: April 22, 2024. Revised: September 15, 2024. Accepted: October 7, 2024. Published: November 6, 2024.
1 Introduction
A lot of investigators are interested in this concept
because it is one of among the most fascinating
subjects in geometric function theory. Analytical
univalent and multivalent functions are holomorphic
and meromorphic functions. This paper is dedicated
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.76
Mustafa I. Hameed, Issa Alkharusi,
Israa A. Ibrahim, Wafaa M. Taha, Ali F. Jameel,
Sundas Nawaz, Mohammad A. Tashtoush
E-ISSN: 2224-2880
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Volume 23, 2024
solely to the study of meromorphic functions, [1],
[2], [3], [4]. The theory of univalent as well as
multivalent functions has emerged as one of the
most important areas to research in the area that
links geometry alongside analysis. The aim of the
research is to become capable to look into a new
class of multivalent functions called an unusual
linear operator and begin studying the new linear
operator with the aid of the product, which stands
for the basic higher-order products resulting from
subordination for various functions, [4], [5]. Using
the universal the characteristics of the broader lead
operator as well as the hypergeometric function,
various solutions are going to be obtained as a more
complex product of deviations and subordination
inside the open unit disk. Very intriguing findings
are found regarding harmonious multivalent
functions that are determined by operators with
differential properties. The idea of shift
subordination is connected to the study of the
univalent function subclass, [6], [7]. We examined
some consequences of both subordination and
Superordination, including a class devoted to the
field of univalent meromorphic functions on a large
unit disc.
The goal of the paper is to show some
applications of the notion of differential
subordination to subclasses of univalent functions
that have certain convolutions as operators. The
aforementioned paper's primary goals are to explore
the dependence principle and introduce an
additional subset over polyvalent functions via an
additional operator associated with higher-order
derivative products.
The following are some of the most significant
scientific applications that can be researched:
communications (hide function), electrostatic fields,
magnetism, heat flow, fluid mechanics, and gravity.
If the functions are analytic functions in
be an open unit disc in as
well as
. The class of analytic
functions in is denoted by , as well as the
subclass of of the kind represented by
where
.
A function is subordinate to and is called
superordinate to if a function is available
the Schwarz
function, so
In case, we compose a letter where
. There would then be equivalency if h is
univalent in.
If is one of , then
The differential subordinations approach was
proposed by Miller and Mocanu in 1978, along with
the theory started to take shape in 1981.
Allow and is
univalent. If function is analytic with
and broad generalizations of inference:
In addition to satisfies the differential subordination:
If are univalent in
in addition to satisfies the differential
subordination:
by Eq. (3),
An explanation of the paper's structure is
provided below. We cover some fundamental
definitions in complex analysis in section two, along
with some examples and fundamental ideas in
geometric function theory. In the third section, we
discuss results concerning the linear operator.
2 Definitions and Lemmas
In this section, the definitions you need in the
theorems are mentioned, a new operator is
mentioned, and the special cases obtained are
mentioned.
Definition 2.1: [8] Allow the set of analytic as
well as injective functions on
so that
In addition to
such that for
is
subclass of for which .
Definition 2.2: [9] Allow be a set in and this
functions that satisfy the
admissibility condition in addition
to
with
. The class of admissible functions
and
for
, and
Definition 2.3: [10] The class of admissible
functions consist of functions
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.76
Mustafa I. Hameed, Issa Alkharusi,
Israa A. Ibrahim, Wafaa M. Taha, Ali F. Jameel,
Sundas Nawaz, Mohammad A. Tashtoush
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Volume 23, 2024
that satisfy the admissibility condition
in addition to
,
so that
. Let be a set in ,
and be a positive integer. In particular
0 as well as , then
and .
In this case, we set , and in
the special case when , the class is simply
denoted by .
Definition 2.4: [11] A derivative operator , which
we apply to the function in cases where
and is a fixed positive natural
number
By Eq. (5), we obtain
Definition 2.5: [12] The generalized derivative
operator
for is described by
Where .
Clearly visible is the fact that
and
and,
We are able to confirm that:
Definition 2.6: By use the Hadamard product of the
derivative operator
and the generalized
operator
and for the operator
is defined
and
so that
By Eq. (9),
As an alternative, consider a few special cases
involving operator
.
1. The generalized derived operator
is present if
by [13], [14], [15].
a. The Srivastava-Attiya derivative operator [16]
introduces
, which is what happens if
and
reduces to
.
b. The Salagean derivative operator [17], [18],
[19] introduces
, which is what happens if
and
reduces to
.
c. The Generalized Salagean derivative operator
introduced by [20] introduces
, which is
what happens if and
reduces to
.
2. The derivative operator
is present if
by [21].
a. [22] introduces
, which is what happens if
and
reduces to
.
b. [23], [24], [25] introduces, which is what
happens if and
reduces
to .
Lemma 2.1: [26] If with
and is univalent in then
Implies .
Lemma 2.2: [26] Let wtih .
If the analytic function
, satisfies the following inclusion relationship
Then
3 Findings Related to the Linear
Operator
In the field of computational science, an operator
which satisfies specific requirements, like having a
closed a subspace and adhering to particular
equations, and keeps linear combinations is called a
linear operator
. Results related to the linear
operator were obtained and are discussed in this
section.
Definition 3.1: Allow
and class of admissible functions consist of
this functions in addition to satisfy
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DOI: 10.37394/23206.2024.23.76
Mustafa I. Hameed, Issa Alkharusi,
Israa A. Ibrahim, Wafaa M. Taha, Ali F. Jameel,
Sundas Nawaz, Mohammad A. Tashtoush
E-ISSN: 2224-2880
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Volume 23, 2024
the admissibility condition , so
that:
and
Where
.
Theorem 3.1: If satisfies
hence,
Proof. By Eq. (5) in addition to Eq. (6),
and define
Hence,
In light of that
Describe the conversion of to by
Let
By use of Lemma 2.1 and Eqs. (16), (17) and (18),
from Eq. (20), we get:
and by Eq. (13), obtaining
See that
since Lemma 2.2. and
Theorem 3.2: If andsatisfies
Therefore,
The following is an expansion of Theorem 3.2
to include the situation where is behavior on
is unknown. For the following result, the best
dominant of the differential subordination is
required Eq. (23).
Theorem 3.3: If , and allow
be univalent in in addition to with
and let where
satisfies
then
Proof: By use Theorem 3.1.,
the conclusion that can now be drawn from the
subsequent subordination relationship
Definition 3.2. If class of admissible functions
of functions in addition to
satisfy the admissibility condition:
so that be a set in , and
and .
Corollary 3.1. If and satisfies
so that Then
.
Proof: By using Theorem 3.1., obtaining
Hence,
, so that .
Corollary 3.2. Let and
satisfy inclusion relationship
so that Then
4 Conclusions
In the present work, some applications of the notion
of different subordination to subclasses of univalent
functions which employ specific transformations as
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DOI: 10.37394/23206.2024.23.76
Mustafa I. Hameed, Issa Alkharusi,
Israa A. Ibrahim, Wafaa M. Taha, Ali F. Jameel,
Sundas Nawaz, Mohammad A. Tashtoush
E-ISSN: 2224-2880
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Volume 23, 2024
operators are found. The geometric properties of
these kinds of functions were studied including,
among other things, convexity radii, starlikeness,
distortion theorem, as well as coefficient bounds.
We investigated both the essential operator as well
as extreme points. Here, we investigated some
characteristics related to the differences in
subordination of analytical univalent functions in .
We also used the characteristics of the more general
result operator to determine some facets of
subordination as well as Superordination. Utilizing a
broader hypergeometric function and the convexity
operator, several results on subordination in .
Applying the principle of dependence and
introducing a new subset of polyvalent functions via
an additional operator associated with higher order
derivative products. The applications of science that
are most relevant to study are fluid mechanics,
communications (hide function), magnetism,
gravity, electrostatic fields, and heat flow.
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Israa A. Ibrahim, Wafaa M. Taha, Ali F. Jameel,
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Source of funding
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.76
Mustafa I. Hameed, Issa Alkharusi,
Israa A. Ibrahim, Wafaa M. Taha, Ali F. Jameel,
Sundas Nawaz, Mohammad A. Tashtoush
E-ISSN: 2224-2880
744
Volume 23, 2024
