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12 pages, 671 KiB

Open AccessArticle

Inequalities of a Class of Analytic Functions Involving Multiplicative Derivative

by Kadhavoor R. Karthikeyan, Daniel Breaz, Gangadharan Murugusundaramoorthy and Ganapathi Thirupathi

Mathematics 2025, 13(10), 1606; https://doi.org/10.3390/math13101606 - 14 May 2025

Viewed by 381

Using the concepts of multiplicative calculus and subordination of analytic functions, we define a new class of starlike bi-univalent functions based on a symmetric operator, which involved the three parameter Mittag-Leffler function. Estimates for the initial coefficients and Fekete–Szegő inequalities of the defined [...] Read more.

Using the concepts of multiplicative calculus and subordination of analytic functions, we define a new class of starlike bi-univalent functions based on a symmetric operator, which involved the three parameter Mittag-Leffler function. Estimates for the initial coefficients and Fekete–Szegő inequalities of the defined function classes are determined. Moreover, special cases of the classes have been discussed and stated as corollaries, which have not been discussed previously. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

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12 pages, 1062 KiB

Open AccessArticle

Subfamilies of Bi-Univalent Functions Associated with the Imaginary Error Function and Subordinate to Jacobi Polynomials

by Ala Amourah, Basem Frasin, Jamal Salah and Feras Yousef

Symmetry 2025, 17(2), 157; https://doi.org/10.3390/sym17020157 - 22 Jan 2025

Cited by 8 | Viewed by 781

Abstract

Numerous researchers have extensively studied various subfamilies of the bi-univalent function family utilizing special functions. In this paper, we introduce and investigate a new subfamily of bi-univalent functions, which is defined on the symmetric domain. This subfamily is connected to the Jacobi polynomial [...] Read more.

Numerous researchers have extensively studied various subfamilies of the bi-univalent function family utilizing special functions. In this paper, we introduce and investigate a new subfamily of bi-univalent functions, which is defined on the symmetric domain. This subfamily is connected to the Jacobi polynomial through the imaginary error function. We derive the initial coefficients of the Maclaurin series for functions in this subfamily, and analyze the Fekete–Szegő inequality for these functions. Additionally, by specializing the parameters in our main results, we deduce several new and significant findings. Full article

(This article belongs to the Section Mathematics)

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14 pages, 367 KiB

Open AccessArticle

Subclasses of Bi-Univalent Functions Connected with Caputo-Type Fractional Derivatives Based upon Lucas Polynomial

by Kholood M. Alsager, Gangadharan Murugusundaramoorthy, Daniel Breaz and Sheza M. El-Deeb

Fractal Fract. 2024, 8(8), 452; https://doi.org/10.3390/fractalfract8080452 - 31 Jul 2024

Cited by 1 | Viewed by 1146

Abstract

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in these subclasses. [...] Read more.

In the current paper, we introduce new subclasses of analytic and bi-univalent functions involving Caputo-type fractional derivatives subordinating to the Lucas polynomial. Furthermore, we find non-sharp estimates on the first two Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in these subclasses. Using the values of

|a_{2}|

and

|a_{3}|

, we determined Fekete–Szegő inequality for functions in these subclasses. Moreover, we pointed out some more subclasses by fixing the parameters involved in Lucas polynomial and stated the estimates and Fekete–Szegő inequality results without proof. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

14 pages, 322 KiB

Open AccessArticle

Coefficient Functionals of Sakaguchi-Type Starlike Functions Involving Caputo-Type Fractional Derivatives Subordinated to the Three-Leaf Function

by Kholood M. Alsager, Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy and Daniel Breaz

Mathematics 2024, 12(14), 2273; https://doi.org/10.3390/math12142273 - 20 Jul 2024

Viewed by 1166

Abstract

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric [...] Read more.

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric points, this article aims to investigate the first three initial coefficient estimates, the bounds for various problems such as Fekete–Szegő inequality, and the Zalcman inequalities, by subordinating to the function of the three leaves domain. Fekete–Szegő-type inequalities and initial coefficients for functions of the form

H^{- 1}

and

\frac{ζ}{H (ζ)}

and

\frac{1}{2} log (\frac{H (ζ)}{ζ})

connected to the three leaves functions are also discussed. Full article

(This article belongs to the Special Issue Geometric Function Theory and Applications―Festschrift for Grigore-Stefan Salagean's 75th Birthday)

19 pages, 521 KiB

Open AccessArticle

Subclasses of Analytic Functions Subordinated to the Four-Leaf Function

by Saravanan Gunasekar, Baskaran Sudharsanan, Musthafa Ibrahim and Teodor Bulboacă

Axioms 2024, 13(3), 155; https://doi.org/10.3390/axioms13030155 - 27 Feb 2024

Cited by 5 | Viewed by 1998

Abstract

The purpose of this research is to unify and extend the study of the well-known concept of coefficient estimates for some subclasses of analytic functions. We define the new subclass

A_{4}^{r, s}

of analytic functions related to the four-leaf domain, [...] Read more.

The purpose of this research is to unify and extend the study of the well-known concept of coefficient estimates for some subclasses of analytic functions. We define the new subclass

A_{4}^{r, s}

of analytic functions related to the four-leaf domain, to increase the adaptability of our investigation. The initial findings are the bound estimates for the coefficients

| a_{n} |

,

n = 2, 3, 4, 5

, among which the bound of

| a_{2} |

is sharp. Also, we include the sharp-function illustration. Additionally, we obtain the upper-bound estimate for the second Hankel determinant for this subclass as well as those for the Fekete–Szegő functional. Finally, for these subclasses, we provide an estimation of the Krushkal inequality for the function class

A_{4}^{r, s}

. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

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13 pages, 1068 KiB

Open AccessArticle

Properties of a Class of Analytic Functions Influenced by Multiplicative Calculus

by Kadhavoor R. Karthikeyan and Gangadharan Murugusundaramoorthy

Fractal Fract. 2024, 8(3), 131; https://doi.org/10.3390/fractalfract8030131 - 23 Feb 2024

Cited by 8 | Viewed by 1732

Abstract

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of [...] Read more.

Motivated by the notion of multiplicative calculus, more precisely multiplicative derivatives, we used the concept of subordination to create a new class of starlike functions. Because we attempted to operate within the existing framework of the design of analytic functions, a number of restrictions, which are in fact strong constraints, have been placed. We redefined our new class of functions using the three-parameter Mittag–Leffler function (Srivastava–Tomovski generalization of the Mittag–Leffler function), in order to increase the study’s adaptability. Coefficient estimates and their Fekete-Szegő inequalities are our main results. We have included a couple of examples to show the closure and inclusion properties of our defined class. Further, interesting bounds of logarithmic coefficients and their corresponding Fekete–Szegő functionals have also been obtained. Full article

(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)

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13 pages, 316 KiB

Open AccessArticle

Coefficient Bounds for Two Subclasses of Analytic Functions Involving a Limacon-Shaped Domain

by Daniel Breaz, Trailokya Panigrahi, Sheza M. El-Deeb, Eureka Pattnayak and Srikandan Sivasubramanian

Symmetry 2024, 16(2), 183; https://doi.org/10.3390/sym16020183 - 3 Feb 2024

Cited by 4 | Viewed by 1556

Abstract

In the current exploration, we defined new subclasses of analytic functions, namely

R_{l i m} (l, ν)

and

C_{l i m} (l, ν)

, defined by subordination linked with a Limacon-shaped domain. We found a [...] Read more.

In the current exploration, we defined new subclasses of analytic functions, namely

R_{l i m} (l, ν)

and

C_{l i m} (l, ν)

, defined by subordination linked with a Limacon-shaped domain. We found a few initial coefficient bounds and Fekete–Szegő inequalities for the functions in the above-stated new classes. The corresponding results have been derived for the function

h^{- 1}

. Additionally, we discuss the Poisson distribution as an application of our consequences. Full article

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

13 pages, 415 KiB

Open AccessArticle

Fekete–Szegő and Zalcman Functional Estimates for Subclasses of Alpha-Convex Functions Related to Trigonometric Functions

by Krishnan Marimuthu, Uma Jayaraman and Teodor Bulboacă

Mathematics 2024, 12(2), 234; https://doi.org/10.3390/math12020234 - 11 Jan 2024

Cited by 3 | Viewed by 1276

Abstract

In this study, we introduce the new subclasses,

M_{α} (sin)

and

M_{α} (cos)

, of

α

-convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of [...] Read more.

In this study, we introduce the new subclasses,

M_{α} (sin)

and

M_{α} (cos)

, of

α

-convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of the Zalcman functional for the class

M_{α} (cos)

for the case

n = 3

, proving that the Zalcman conjecture holds for this value of n. Moreover, the problem of the Fekete–Szegő functional estimate for these classes is studied. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory)

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14 pages, 444 KiB

Open AccessArticle

Coefficient Inequalities for q-Convex Functions with Respect to q-Analogue of the Exponential Function

by Majid Khan, Nazar Khan, Ferdous M. O. Tawfiq and Jong-Suk Ro

Axioms 2023, 12(12), 1130; https://doi.org/10.3390/axioms12121130 - 15 Dec 2023

Cited by 2 | Viewed by 1680

Abstract

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions [...] Read more.

In mathematical analysis, the q-analogue of a function refers to a modified version of the function that is derived from q-series expansions. This paper is focused on the q-analogue of the exponential function and investigates a class of convex functions associated with it. The main objective is to derive precise inequalities that bound the coefficients of these convex functions. In this research, the initial coefficient bounds, Fekete–Szegő problem, second and third Hankel determinant have been determined. These coefficient bounds provide valuable information about the behavior and properties of the functions within the considered class. Full article

(This article belongs to the Special Issue Honorary Special Issue Dedicated to Prof. Anatolii Asirovich Gol’dberg (1930–2008))

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15 pages, 336 KiB

Open AccessArticle

Certain Quantum Operator Related to Generalized Mittag–Leffler Function

by Mansour F. Yassen and Adel A. Attiya

Mathematics 2023, 11(24), 4963; https://doi.org/10.3390/math11244963 - 15 Dec 2023

Cited by 1 | Viewed by 1171

Abstract

In this paper, we present a novel class of analytic functions in the form

h (z) = z^{p} + \sum_{k = p + 1}^{\infty} a_{k} z^{k}

in the unit disk. These functions establish a connection between [...] Read more.

In this paper, we present a novel class of analytic functions in the form

h (z) = z^{p} + \sum_{k = p + 1}^{\infty} a_{k} z^{k}

in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by

ℵ_{q, p}^{n} (_{L}, a, b)

and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory)

16 pages, 323 KiB

Open AccessArticle

New Subclass of Close-to-Convex Functions Defined by Quantum Difference Operator and Related to Generalized Janowski Function

by Suha B. Al-Shaikh, Mohammad Faisal Khan, Mustafa Kamal and Naeem Ahmad

Symmetry 2023, 15(11), 1974; https://doi.org/10.3390/sym15111974 - 25 Oct 2023

Cited by 2 | Viewed by 1329

Abstract

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex [...] Read more.

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex functions connected with generalized Janowski functions. We prove the necessary and sufficient conditions for functions that belong to newly defined classes, including the inclusion relations and estimations of the coefficients. The Fekete–Szegő problem for a more general class is also discussed. The results of this investigation expand upon those of the previous study. Full article

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

22 pages, 357 KiB

Open AccessArticle

Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain

by Isra Al-Shbeil, Muhammad Imran Faisal, Muhammad Arif, Muhammad Abbas and Reem K. Alhefthi

Mathematics 2023, 11(17), 3664; https://doi.org/10.3390/math11173664 - 25 Aug 2023

Cited by 1 | Viewed by 1437

Abstract

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these [...] Read more.

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory)

18 pages, 336 KiB

Open AccessArticle

Certain New Applications of Faber Polynomial Expansion for a New Class of bi-Univalent Functions Associated with Symmetric q-Calculus

by Chetan Swarup

Symmetry 2023, 15(7), 1407; https://doi.org/10.3390/sym15071407 - 13 Jul 2023

Viewed by 1099

Abstract

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order

γ

. We used the Faber polynomial expansion method to determine the upper bounds for the [...] Read more.

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order

γ

. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order

γ

. For the functions in the newly specified class, we further obtained coefficient bounds

|ρ_{2}|

and the Fekete–Szegő problem

|ρ_{3} - ρ_{2}^{2}|

, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean q-differential operator using the Faber polynomial expansion technique. The findings in this paper generalize those from previous studies. Full article

16 pages, 472 KiB

Open AccessArticle

Initial Coefficient Bounds for Bi-Univalent Functions Related to Gregory Coefficients

by Gangadharan Murugusundaramoorthy, Kaliappan Vijaya and Teodor Bulboacă

Mathematics 2023, 11(13), 2857; https://doi.org/10.3390/math11132857 - 26 Jun 2023

Cited by 14 | Viewed by 1683

Abstract

In this article we introduce three new subclasses of the class of bi-univalent functions

Σ

, namely

{HG}_{Σ}

,

{GM}_{Σ} (μ)

and

G_{Σ} (λ)

, by using the subordinations with the functions whose coefficients are Gregory [...] Read more.

In this article we introduce three new subclasses of the class of bi-univalent functions

Σ

, namely

{HG}_{Σ}

,

{GM}_{Σ} (μ)

and

G_{Σ} (λ)

, by using the subordinations with the functions whose coefficients are Gregory numbers. First, we evidence that these classes are not empty, i.e., they contain other functions besides the identity one. For functions in each of these three bi-univalent function classes, we investigate the estimates

|a_{2}|

and

|a_{3}|

of the Taylor–Maclaurin coefficients and Fekete–Szegő functional problems. The main results are followed by some particular cases, and the novelty of the characterizations and the proofs may lead to further studies of such types of similarly defined subclasses of analytic bi-univalent functions. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)

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16 pages, 371 KiB

Open AccessArticle

Jackson Differential Operator Associated with Generalized Mittag–Leffler Function

by Adel A. Attiya, Mansour F. Yassen and Abdelhamid Albaid

Fractal Fract. 2023, 7(5), 362; https://doi.org/10.3390/fractalfract7050362 - 28 Apr 2023

Cited by 4 | Viewed by 1218

Abstract

Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which [...] Read more.

Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which relates to both the generalized Mittag–Leffler function and the Jackson differential operator. By using a differential subordination virtue, we obtain some important properties such as coefficient bounds and the Fekete–Szegő problem. Some results that represent special cases of this family that have been studied before are also highlighted. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

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