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17 pages, 307 KB

Open AccessArticle

Generalization of the Rafid Operator and Its Symmetric Role in Meromorphic Function Theory with Electrostatic Applications

by Aya F. Elkhatib, Atef F. Hashem, Adela O. Mostafa and Mohammed M. Tharwat

Symmetry 2025, 17(11), 1837; https://doi.org/10.3390/sym17111837 - 2 Nov 2025

Viewed by 399

Abstract

This study introduces a new integral operator

I_{p, μ}^{δ}

that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses:

Σ_{p}^{+} (δ, μ, α)

, consisting [...] Read more.

This study introduces a new integral operator

I_{p, μ}^{δ}

that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses:

Σ_{p}^{+} (δ, μ, α)

, consisting of functions with nonnegative coefficients, and

Σ_{p}^{+} (δ, μ, α, c)

, which further fixes the second positive coefficient. For these classes, we establish a necessary and sufficient coefficient condition, which serves as the foundation for deriving a set of sharp results. These include accurate coefficient bounds, distortion theorems for functions and derivatives, and radii of starlikeness and convexity of a specific order. Furthermore, we demonstrate the closure property of the class

Σ_{p}^{+} (δ, μ, α, c)

, identify its extreme points, and then construct a neighborhood theorem. All the findings presented in this paper are sharp. To demonstrate the practical utility of our symmetric operator paradigm, we apply it to a canonical fractional electrodynamics problem. We demonstrate how sharp distortion theorems establish rigorous, time-invariant upper bounds for a solitary electrostatic potential and its accompanying electric field, resulting in a mathematically guaranteed safety buffer against dielectric breakdown. This study develops a symmetric and consistent approach to investigating the geometric characteristics of meromorphic multivalent functions and their applications in physical models. Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

17 pages, 273 KB

Open AccessArticle

A Class of Meromorphic Multivalent Functions with Negative Coefficients Defined by a Ruscheweyh-Type Operator

by Isabel Marrero

Axioms 2025, 14(4), 284; https://doi.org/10.3390/axioms14040284 - 9 Apr 2025

Cited by 2 | Viewed by 817

Abstract

We introduce and systematically study a new class

k^{λ, p} (α, β)

of meromorphic p-valent functions defined by means of the Ruscheweyh-type operator

D_{*}^{λ, p}

, where

p \in N

, [...] Read more.

We introduce and systematically study a new class

k^{λ, p} (α, β)

of meromorphic p-valent functions defined by means of the Ruscheweyh-type operator

D_{*}^{λ, p}

, where

p \in N

,

λ > - p

,

0 \leq α < 1

, and

β > 0

. Membership in this class is characterized through coefficient estimates. Also investigated are growth, distortion, stability under convex combinations, radii of starlikeness and convexity of order

ρ

(0 \leq ρ < 1)

, convolution, the action of an integral operator of Bernardi–Libera–Livingston type, and neighborhoods. Full article

(This article belongs to the Section Mathematical Analysis)

11 pages, 252 KB

Open AccessArticle

by Luminiţa-Ioana Cotîrlă and Elisabeta-Alina Totoi

Symmetry 2024, 16(12), 1604; https://doi.org/10.3390/sym16121604 - 2 Dec 2024

Viewed by 1059

Abstract

We define new classes of meromorphic p-valent convex functions, respectively, meromorphic close-to-convex functions, by using an extension of Wanas operator in order to study the preservation properties of these classes, when a well-known integral operator is used. We find the conditions which allow [...] Read more.

We define new classes of meromorphic p-valent convex functions, respectively, meromorphic close-to-convex functions, by using an extension of Wanas operator in order to study the preservation properties of these classes, when a well-known integral operator is used. We find the conditions which allow this operator to preserve the classes mentioned above, and we will remark the symmetry between these classes. Full article

(This article belongs to the Special Issue Geometric Function Theory and Special Functions II)

16 pages, 333 KB

Open AccessArticle

Binomial Series-Confluent Hypergeometric Distribution and Its Applications on Subclasses of Multivalent Functions

by Ibtisam Aldawish, Sheza M. El-Deeb and Gangadharan Murugusundaramoorthy

Symmetry 2023, 15(12), 2186; https://doi.org/10.3390/sym15122186 - 11 Dec 2023

Cited by 3 | Viewed by 1618

Abstract

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this [...] Read more.

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

12 pages, 274 KB

Open AccessArticle

Integral Operators Applied to Classes of Convex and Close-to-Convex Meromorphic p-Valent Functions

by Elisabeta-Alina Totoi and Luminita-Ioana Cotirla

Symmetry 2023, 15(11), 2079; https://doi.org/10.3390/sym15112079 - 17 Nov 2023

Viewed by 1171

Abstract

We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this operator has to meet in order to preserve convex meromorphic functions. We know that convexity [...] Read more.

We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this operator has to meet in order to preserve convex meromorphic functions. We know that convexity has great impact in the industry, linear and non-linear programming problems, and optimization. Some lemmas and remarks helping us to obtain complex functions with positive real parts are also given. Full article

(This article belongs to the Special Issue Geometric Function Theory and Special Functions II)

19 pages, 384 KB

Open AccessArticle

Subclasses of p-Valent κ-Uniformly Convex and Starlike Functions Defined by the q-Derivative Operator

by Ekram E. Ali, Hari M. Srivastava and Abeer M. Albalahi

Mathematics 2023, 11(11), 2578; https://doi.org/10.3390/math11112578 - 4 Jun 2023

Cited by 12 | Viewed by 2117

Abstract

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are [...] Read more.

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q-calculus in order to introduce the q-derivative operator

S_{η, p, q}^{n, m}

. Secondly, by means of this q-derivative operator, we define an interesting subclass

T ℵ_{λ, p}^{n, m} (η, α, κ)

of the class of normalized analytic and multivalent (or p-valent) functions in the open unit disk

U

. This p-valent analytic function class is associated with the class

κ

-

UCV

of

κ

-uniformly convex functions and the class

κ

-

UST

of

κ

-uniformly starlike functions in

U

. For functions belonging to the normalized analytic and multivalent (or p-valent) function class

T ℵ_{λ, p}^{n, m} (η, α, κ)

, we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein. Full article

(This article belongs to the Special Issue New Trends in Complex Analysis Research, 2nd Edition)

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13 pages, 308 KB

Open AccessArticle

Subclasses of p-Valent Functions Associated with Linear q-Differential Borel Operator

by Adriana Cătaş, Emilia-Rodica Borşa and Sheza M. El-Deeb

Mathematics 2023, 11(7), 1742; https://doi.org/10.3390/math11071742 - 5 Apr 2023

Viewed by 1634

Abstract

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the [...] Read more.

The aim of the present paper is to introduce and study some new subclasses of p-valent functions by making use of a linear q-differential Borel operator.We also deduce some properties, such as inclusion relationships of the newly introduced classes and the integral operator

J_{μ, p}

. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications)

15 pages, 328 KB

Open AccessArticle

A Class of Janowski-Type (p,q)-Convex Harmonic Functions Involving a Generalized q-Mittag–Leffler Function

by Sarem H. Hadi, Maslina Darus and Alina Alb Lupaş

Axioms 2023, 12(2), 190; https://doi.org/10.3390/axioms12020190 - 11 Feb 2023

Cited by 10 | Viewed by 2304

Abstract

This research aims to present a linear operator

L_{p, q}^{ρ, σ, μ} f

utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic

(p, q)

-convex functions [...] Read more.

This research aims to present a linear operator

L_{p, q}^{ρ, σ, μ} f

utilizing the q-Mittag–Leffler function. Then, we introduce the subclass of harmonic

(p, q)

-convex functions

{HT}_{p, q} (ϑ, W, V)

related to the Janowski function. For the harmonic p-valent functions f class, we investigate the harmonic geometric properties, such as coefficient estimates, convex linear combination, extreme points, and Hadamard product. Finally, the closure property is derived using the subclass

{HT}_{p, q} (ϑ, W, V)

under the q-Bernardi integral operator. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems with Applications)

14 pages, 298 KB

Open AccessArticle

Application of a Multiplier Transformation to Libera Integral Operator Associated with Generalized Distribution

by Jamiu Olusegun Hamzat, Abiodun Tinuoye Oladipo and Georgia Irina Oros

Symmetry 2022, 14(9), 1934; https://doi.org/10.3390/sym14091934 - 16 Sep 2022

Cited by 2 | Viewed by 1729

Abstract

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function

f_{s} (z)

is defined as involving an analytic [...] Read more.

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function

f_{s} (z)

is defined as involving an analytic p-valent function and generalized distribution expressed in terms of analytic p-valent functions. Neighborhood properties for functions

f_{s} (z)

are established. Further, by applying a previously introduced linear transformation to

f_{s} (z)

and using an extended Libera integral operator, a new generalized Libera-type operator is defined. Moreover, using the same linear transformation, subclasses of starlike, convex, close-to-convex and spiralike functions are defined and investigated in order to obtain geometrical properties that characterize the new generalized Libera-type operator. Symmetry properties are due to the involvement of the Libera integral operator and convolution transform. Full article

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

9 pages, 270 KB

Open AccessArticle

Hadamard Product of Certain Multivalent Analytic Functions with Positive Real Parts

by Abdel Moneim Y. Lashin and Mohamed K. Aouf

Mathematics 2022, 10(9), 1506; https://doi.org/10.3390/math10091506 - 1 May 2022

Cited by 2 | Viewed by 2122

Abstract

This paper aims to provide sufficient conditions for starlikeness and convexity of Hadamard product (convolution) of certain multivalent analytic functions with positive real parts. Moreover, the starlikeness conditions for a certain integral operator and other convolution results are also considered. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory)

17 pages, 312 KB

Open AccessArticle

Applications of Certain p-Valently Analytic Functions

by Georgia Irina Oros, Gheorghe Oros and Shigeyoshi Owa

Mathematics 2022, 10(6), 910; https://doi.org/10.3390/math10060910 - 12 Mar 2022

Cited by 5 | Viewed by 2790

Abstract

In this paper, a new operator

D^{s} f

, with s a real number, is defined considering functions that belong to the known class of p-valent analytic functions in the open unit disk U. Applying this operator, a new subclass [...] Read more.

In this paper, a new operator

D^{s} f

, with s a real number, is defined considering functions that belong to the known class of p-valent analytic functions in the open unit disk U. Applying this operator, a new subclass of p-valently analytic functions is introduced and some interesting subordination- and coefficient-related properties of the functions in this class are discussed. It is also shown that for certain values given to the parameters involved in the definition of the class, p-valently starlike and p-valently convex functions of certain orders can be obtained, respectively. Examples are also given as applications of the newly proven results. Full article

16 pages, 297 KB

Open AccessArticle

A Certain Subclass of Multivalent Analytic Functions Defined by the q-Difference Operator Related to the Janowski Functions

by Bo Wang, Rekha Srivastava and Jin-Lin Liu

Mathematics 2021, 9(14), 1706; https://doi.org/10.3390/math9141706 - 20 Jul 2021

Cited by 15 | Viewed by 2292

Abstract

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of [...] Read more.

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of convexity and starlikeness, closure theorems and partial sums, are discussed in this paper. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

14 pages, 303 KB

Open AccessArticle

A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences

by Qiuxia Hu, Hari M. Srivastava, Bakhtiar Ahmad, Nazar Khan, Muhammad Ghaffar Khan, Wali Khan Mashwani and Bilal Khan

Symmetry 2021, 13(7), 1275; https://doi.org/10.3390/sym13071275 - 16 Jul 2021

Cited by 30 | Viewed by 2875

Abstract

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class

A_{p}

, where class

A_{p}

is invariant (or symmetric) under rotations. The well-known class [...] Read more.

In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class

A_{p}

, where class

A_{p}

is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the

(p, q)

-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter

p

is obviously unnecessary. Full article

(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)

9 pages, 224 KB

Open AccessArticle

The Order of Strongly Starlikeness of the Generalized α-Convex Functions

by Yuan Yuan, Rekha Srivastava and Jin-Lin Liu

Symmetry 2019, 11(1), 76; https://doi.org/10.3390/sym11010076 - 11 Jan 2019

Cited by 1 | Viewed by 5246

Abstract

We consider the order of the strongly-starlikeness of the generalized

α

-convex functions. Some sufficient conditions for functions to be p-valently strongly-starlike are given. Full article

(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

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