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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 1165

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)

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 1275

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 1674

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 1325

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 1679

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 1213

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)

15 pages, 315 KiB

Open AccessArticle

New Applications of Faber Polynomial Expansion for Analytical Bi-Close-to-Convex Functions Defined by Using q-Calculus

by Ridong Wang, Manoj Singh, Shahid Khan, Huo Tang, Mohammad Faisal Khan and Mustafa Kamal

Mathematics 2023, 11(5), 1217; https://doi.org/10.3390/math11051217 - 1 Mar 2023

Cited by 3 | Viewed by 1679

Abstract

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the [...] Read more.

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the unpredictable behaviour of initial coefficients

|a_{2}|

,

|a_{3}|

and investigate the Fekete–Szegő problem

|a_{3} - a_{2}^{2}|

for the subclasses of bi-close-to-convex functions. To highlight the connections between existing knowledge and new research, certain known and unknown corollaries are also highlighted. Full article

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

7 pages, 269 KiB

Open AccessArticle

Applications of Beta Negative Binomial Distribution and Laguerre Polynomials on Ozaki Bi-Close-to-Convex Functions

by Isra Al-Shbeil, Abbas Kareem Wanas, Afis Saliu and Adriana Cătaş

Axioms 2022, 11(9), 451; https://doi.org/10.3390/axioms11090451 - 2 Sep 2022

Cited by 17 | Viewed by 2242

Abstract

In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family

F_{Σ} (δ, η, λ, θ; h)

of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. [...] Read more.

In the present paper, due to beta negative binomial distribution series and Laguerre polynomials, we investigate a new family

F_{Σ} (δ, η, λ, θ; h)

of normalized holomorphic and bi-univalent functions associated with Ozaki close-to-convex functions. We provide estimates on the initial Taylor–Maclaurin coefficients and discuss Fekete–Szegő type inequality for functions in this family. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory)

17 pages, 999 KiB

Open AccessArticle

Classes of Multivalent Spirallike Functions Associated with Symmetric Regions

by Luminiţa-Ioana Cotîrlǎ and Kadhavoor R. Karthikeyan

Symmetry 2022, 14(8), 1598; https://doi.org/10.3390/sym14081598 - 3 Aug 2022

Cited by 3 | Viewed by 1736

Abstract

We define a function to unify the well-known class of Janowski functions with a class of spirallike functions of reciprocal order. We focus on the impact of defined function on various conic regions which are symmetric with respect to the real axis. Further, [...] Read more.

We define a function to unify the well-known class of Janowski functions with a class of spirallike functions of reciprocal order. We focus on the impact of defined function on various conic regions which are symmetric with respect to the real axis. Further, we have defined a new subclass of multivalent functions of complex order subordinate to the extended Janowski function. This work bridges the studies of various subclasses of spirallike functions and extends well-known results. Interesting properties have been obtained for the defined function class. Several consequences of our main results have been pointed out. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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13 pages, 331 KiB

Open AccessArticle

A Differential Operator Associated with q-Raina Function

by Adel A. Attiya, Rabha W. Ibrahim, Abeer M. Albalahi, Ekram E. Ali and Teodor Bulboacă

Symmetry 2022, 14(8), 1518; https://doi.org/10.3390/sym14081518 - 25 Jul 2022

Cited by 9 | Viewed by 1861

Abstract

The topics studied in the geometric function theory of one variable functions are connected with the concept of Symmetry because for some special cases the analytic functions map the open unit disk onto a symmetric domain. Thus, if all the coefficients of the [...] Read more.

The topics studied in the geometric function theory of one variable functions are connected with the concept of Symmetry because for some special cases the analytic functions map the open unit disk onto a symmetric domain. Thus, if all the coefficients of the Taylor expansion at the origin are real numbers, then the image of the open unit disk is a symmetric domain with respect to the real axis. In this paper, we formulate the q-differential operator associated with the q-Raina function using quantum calculus, that is the so-called Jacksons’ calculus. We establish a new subclass of analytic functions in the unit disk by using this newly developed operator. The theory of differential subordination inspired our approach; therefore, we geometrically explore the most popular properties of this new operator: subordination properties, coefficient bounds, and the Fekete-Szegő problem. As special cases, we highlight certain well-known corollaries of our primary findings. Full article

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

8 pages, 281 KiB

Open AccessArticle

An Avant-Garde Construction for Subclasses of Analytic Bi-Univalent Functions

by Feras Yousef, Ala Amourah, Basem Aref Frasin and Teodor Bulboacă

Axioms 2022, 11(6), 267; https://doi.org/10.3390/axioms11060267 - 1 Jun 2022

Cited by 38 | Viewed by 2448

Abstract

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates [...] Read more.

The zero-truncated Poisson distribution is an important and appropriate model for many real-world applications. Here, we exploit the zero-truncated Poisson distribution probabilities to construct a new subclass of analytic bi-univalent functions involving Gegenbauer polynomials. For functions in the constructed class, we explore estimates of Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

, and next, we solve the Fekete–Szegő functional problem. A number of new interesting results are presented to follow upon specializing the parameters involved in our main results. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory)

13 pages, 331 KiB

Open AccessArticle

Subclasses of Yamakawa-Type Bi-Starlike Functions Associated with Gegenbauer Polynomials

by Gangadharan Murugusundaramoorthy and Teodor Bulboacă

Axioms 2022, 11(3), 92; https://doi.org/10.3390/axioms11030092 - 24 Feb 2022

Cited by 13 | Viewed by 2613

Abstract

In this paper, we introduce and investigate new subclasses (Yamakawa-type bi-starlike functions and another class of Lashin, both mentioned in the reference list) of bi-univalent functions defined in the open unit disk, which are associated with the Gegenbauer polynomials and satisfy subordination conditions. [...] Read more.

In this paper, we introduce and investigate new subclasses (Yamakawa-type bi-starlike functions and another class of Lashin, both mentioned in the reference list) of bi-univalent functions defined in the open unit disk, which are associated with the Gegenbauer polynomials and satisfy subordination conditions. Furthermore, we find estimates for the Taylor–Maclaurin coefficients

| a_{2} |

and

| a_{3} |

for functions in these new subclasses. Several known or new consequences of the results are also pointed out. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory)

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