MDPI - Publisher of Open Access Journals

13 pages, 290 KB

Open AccessArticle

Bi-Univalent Function Classes Defined by Imaginary Error Function and Bernoulli Polynomials

by Ibtisam Aldawish, Sondekola Rudra Swamy, Basem Aref Frasin and Supriya Chandrashekharaiah

Axioms 2025, 14(10), 731; https://doi.org/10.3390/axioms14100731 - 27 Sep 2025

Viewed by 179

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in

U =

[...] Read more.

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in

U =

{ς \in C : | ς | < 1}

, characterized by Bernoulli polynomials associated with imaginary error functions. For functions belonging to these subclasses, we establish bounds for their initial coefficients. For these classes, we also tackle the Fekete–Szegö problem. Several new results are also obtained as special cases by specifying certain parameter values in the general findings. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 4th Edition)

14 pages, 301 KB

Open AccessArticle

Coefficient Estimates, the Fekete–Szegö Inequality, and Hankel Determinants for Universally Prestarlike Functions Defined by Fractional Derivative in a Shell-Shaped Region

by Dina Nabil, Georgia Irina Oros, Awatef Shahin and Hanan Darwish

Axioms 2025, 14(9), 711; https://doi.org/10.3390/axioms14090711 - 21 Sep 2025

Viewed by 277

Abstract

In this paper, we introduce and investigate a new subclass

{(R_{ς}^{u})}^{g} (ϕ)

of universally prestarlike generalized functions of order

ς

, where

ς \leq 1

, associated with a shell-shaped region defined by [...] Read more.

In this paper, we introduce and investigate a new subclass

{(R_{ς}^{u})}^{g} (ϕ)

of universally prestarlike generalized functions of order

ς

, where

ς \leq 1

, associated with a shell-shaped region defined by

Λ = C ∖ [1, \infty)

for the present investigation, by utilizing the Srivastava–Owa fractional derivative of order

δ

. Coefficient inequalities for

| a_{2} |

and

| a_{3} |

for functions belonging to the newly introduced class are obtained. Additionally, the Fekete–Szegö inequality is investigated for this class of functions. In order to enhance the coefficient studies for this class, the second Hankel determinant is also evaluated. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 4th Edition)

18 pages, 724 KB

Open AccessArticle

Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the q-Bernoulli Polynomial

by Abbas Kareem Wanas, Qasim Ali Shakir and Adriana Catas

Symmetry 2025, 17(9), 1532; https://doi.org/10.3390/sym17091532 - 13 Sep 2025

Viewed by 434

Abstract

In the present work, we define certain families,

M_{Σ} (μ, Υ, ℷ, q; x)

and

N_{Σ} (μ, Υ, ℷ, q; x)

, of normalized holomorphic and bi-univalent functions associated with Bazilevič [...] Read more.

In the present work, we define certain families,

M_{Σ} (μ, Υ, ℷ, q; x)

and

N_{Σ} (μ, Υ, ℷ, q; x)

, of normalized holomorphic and bi-univalent functions associated with Bazilevič functions and

ℷ

-pseudo functions involving the

q

-Bernoulli polynomial, which is defined by the symmetric nature of quantum calculus in the open unit disk

U

. We determine the upper bounds for the initial symmetry Taylor–Maclaurin coefficients and the Fekete–Szegö-type inequalities of functions in the families we have introduced here. In addition, we indicate certain special cases and consequences for our results. Full article

(This article belongs to the Special Issue Applications of Special Functions in Complex Analysis and Their Symmetries)

► Show Figures

Figure 1

25 pages, 401 KB

Open AccessArticle

Coefficient Bounds for Alpha-Convex Functions Involving the Linear q-Derivative Operator Connected with the Cardioid Domain

by Sudhansu Palei, Madan Mohan Soren and Luminiţa-Ioana Cotîrlǎ

Fractal Fract. 2025, 9(3), 172; https://doi.org/10.3390/fractalfract9030172 - 12 Mar 2025

Cited by 2 | Viewed by 838

Abstract

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator [...] Read more.

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator

S_{μ, δ, q}^{n, m}

and subordination are used in this study to define and construct new classes of

α

-convex functions associated with the cardioid domain. Additionally, this paper explores acute inequality problems for newly defined classes

R_{q}^{α} (a, c, m, L, P),

of

α

-convex functions in the open unit disc

U_{s}

, such as initial coefficient bounds, coefficient inequalities, Fekete–Szegö problems, the second Hankel determinants, and logarithmic coefficients. The results presented in this paper are simple to comprehend and demonstrate how current research relates to earlier research. We found all of the estimates, and they are sharp. Full article

(This article belongs to the Section General Mathematics, Analysis)

11 pages, 257 KB

Open AccessArticle

Comprehensive Subfamilies of Bi-Univalent Functions Defined by Error Function Subordinate to Euler Polynomials

by Tariq Al-Hawary, Basem Frasin and Jamal Salah

Symmetry 2025, 17(2), 256; https://doi.org/10.3390/sym17020256 - 8 Feb 2025

Cited by 1 | Viewed by 722

Abstract

Recently, several researchers have estimated the Maclaurin coefficients, namely

|q_{2}|

and

|q_{3}|,

and the Fekete–Szegö functional problem of functions belonging to some special subfamilies of analytic functions related to certain polynomials, such as Lucas polynomials, Legendrae polynomials, Chebyshev polynomials, and [...] Read more.

Recently, several researchers have estimated the Maclaurin coefficients, namely

|q_{2}|

and

|q_{3}|,

and the Fekete–Szegö functional problem of functions belonging to some special subfamilies of analytic functions related to certain polynomials, such as Lucas polynomials, Legendrae polynomials, Chebyshev polynomials, and others. This study obtains the bounds of coefficients

|q_{2}|

and

|q_{3}|

, and the Fekete–Szegö functional problem for functions belonging to the comprehensive subfamilies

T (ζ, ϵ, δ)

and

J (φ, δ)

of analytic functions in a symmetric domain

U

, using the imaginary error function subordinate to Euler polynomials. After specializing the parameters used in our main results, a number of new special cases are also obtained. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Complex Analysis by the Means of Special Functions)

19 pages, 317 KB

Open AccessArticle

Sharp Second-Order Hankel Determinants Bounds for Alpha-Convex Functions Connected with Modified Sigmoid Functions

by Muhammad Abbas, Reem K. Alhefthi, Daniele Ritelli and Muhammad Arif

Axioms 2024, 13(12), 844; https://doi.org/10.3390/axioms13120844 - 1 Dec 2024

Cited by 3 | Viewed by 1048

Abstract

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a [...] Read more.

The study of the Hankel determinant generated by the Maclaurin series of holomorphic functions belonging to particular classes of normalized univalent functions is one of the most significant problems in geometric function theory. Our goal in this study is first to define a family of alpha-convex functions associated with modified sigmoid functions and then to investigate sharp bounds of initial coefficients, Fekete-Szegö inequality, and second-order Hankel determinants. Moreover, we also examine the logarithmic and inverse coefficients of functions within a defined family regarding recent issues. All of the estimations that were found are sharp. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Related Topics)

17 pages, 528 KB

Open AccessArticle

Applications of a q-Integral Operator to a Certain Class of Analytic Functions Associated with a Symmetric Domain

by Adeel Ahmad, Hanen Louati, Akhter Rasheed, Asad Ali, Saqib Hussain, Shreefa O. Hilali and Afrah Y. Al-Rezami

Symmetry 2024, 16(11), 1443; https://doi.org/10.3390/sym16111443 - 31 Oct 2024

Cited by 1 | Viewed by 1403

Abstract

In this article, our objective is to define and study a new subclass of analytic functions associated with the q-analogue of the sine function, operating in conjunction with a convolution operator. By manipulating the parameter q, we observe that the image [...] Read more.

In this article, our objective is to define and study a new subclass of analytic functions associated with the q-analogue of the sine function, operating in conjunction with a convolution operator. By manipulating the parameter q, we observe that the image of the unit disc under the q-sine function exhibits a visually appealing resemblance to a figure-eight shape that is symmetric about the real axis. Additionally, we investigate some important geometrical problems like necessary and sufficient conditions, coefficient bounds, Fekete-Szegö inequality, and partial sum results for the functions belonging to this newly defined subclass. Full article

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

► Show Figures

Figure 1

13 pages, 284 KB

Open AccessArticle

On the Fekete–Szegö Problem for Certain Classes of (γ,δ)-Starlike and (γ,δ)-Convex Functions Related to Quasi-Subordinations

by Norah Saud Almutairi, Awatef Shahen, Adriana Cătaş and Hanan Darwish

Symmetry 2024, 16(8), 1043; https://doi.org/10.3390/sym16081043 - 14 Aug 2024

Cited by 3 | Viewed by 1200

Abstract

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates

| a_{3} - μ a_{2}^{2} |

for functions belonging to [...] Read more.

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates

| a_{3} - μ a_{2}^{2} |

for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined. Full article

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

19 pages, 310 KB

Open AccessArticle

Fractional Differential Operator Based on Quantum Calculus and Bi-Close-to-Convex Functions

by Zeya Jia, Alina Alb Lupaş, Haifa Bin Jebreen, Georgia Irina Oros, Teodor Bulboacă and Qazi Zahoor Ahmad

Mathematics 2024, 12(13), 2026; https://doi.org/10.3390/math12132026 - 29 Jun 2024

Cited by 1 | Viewed by 1494

Abstract

In this article, we first consider the fractional q-differential operator and the

(λ, q)

-fractional differintegral operator

D_{q}^{λ} : A \to A

. Using the

(λ, q)

-fractional differintegral operator, we define two new subclasses of analytic functions: [...] Read more.

In this article, we first consider the fractional q-differential operator and the

(λ, q)

-fractional differintegral operator

D_{q}^{λ} : A \to A

. Using the

(λ, q)

-fractional differintegral operator, we define two new subclasses of analytic functions: the subclass

S^{*} (q, β, λ)

of starlike functions of order

β

and the class

C_{Σ}^{λ, q} (α)

of bi-close-to-convex functions of order

β

. We explore the results on coefficient inequality and Fekete–Szegö problems for functions belonging to the class

S^{*} (q, β, λ)

. Using the Faber polynomial technique, we derive upper bounds for the nth coefficient of functions in the class of bi-close-to-convex functions of order

β

. We also investigate the erratic behavior of the initial coefficients in the class

C_{Σ}^{λ, q} (α)

of bi-close-to-convex functions. Furthermore, we address some known problems to demonstrate the connection between our new work and existing research. Full article

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

13 pages, 1417 KB

Open AccessArticle

A Bi-Starlike Class in a Leaf-like Domain Defined through Subordination via

q ̧

-Calculus

by Ala Amourah, Abdullah Alsoboh, Daniel Breaz and Sheza M. El-Deeb

Mathematics 2024, 12(11), 1735; https://doi.org/10.3390/math12111735 - 3 Jun 2024

Cited by 13 | Viewed by 1099

Abstract

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with

q

-calculus. [...] Read more.

Bi-univalent functions associated with the leaf-like domain within the open unit disk are investigated and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with

q

-calculus. The class is proved to be not empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on

| α_{2} |

and

| α_{3} |

coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. Full article

(This article belongs to the Special Issue Advances in Complex Analysis and Application)

► Show Figures

Figure 1

15 pages, 659 KB

Open AccessArticle

A Class of Bi-Univalent Functions in a Leaf-Like Domain Defined through Subordination via

q ̧

-Calculus

by Abdullah Alsoboh and Georgia Irina Oros

Mathematics 2024, 12(10), 1594; https://doi.org/10.3390/math12101594 - 20 May 2024

Cited by 14 | Viewed by 1532

Abstract

Bi-univalent functions associated with the leaf-like domain within open unit disks are investigated, and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with

q

-calculus. The [...] Read more.

Bi-univalent functions associated with the leaf-like domain within open unit disks are investigated, and a new subclass is introduced and studied in the research presented here. This is achieved by applying the subordination principle for analytic functions in conjunction with

q

-calculus. The class is proved to not be empty. By proving its existence, generalizations can be given to other sets of functions. In addition, coefficient bounds are examined with a particular focus on

| α_{2} |

and

| α_{3} |

coefficients, and Fekete–Szegö inequalities are estimated for the functions in this new class. To support the conclusions, previous works are cited for confirmation. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

► Show Figures

Figure 1

16 pages, 305 KB

Open AccessArticle

Second Hankel Determinant and Fekete–Szegö Problem for a New Class of Bi-Univalent Functions Involving Euler Polynomials

by Semh Kadhim Gebur and Waggas Galib Atshan

Symmetry 2024, 16(5), 530; https://doi.org/10.3390/sym16050530 - 28 Apr 2024

Cited by 2 | Viewed by 1121

Abstract

Orthogonal polynomials have been widely employed by renowned authors within the context of geometric function theory. This study is driven by prior research and aims to address the —Fekete-Szegö problem. Additionally, we provide bound estimates for the coefficients and an upper bound estimate [...] Read more.

Orthogonal polynomials have been widely employed by renowned authors within the context of geometric function theory. This study is driven by prior research and aims to address the —Fekete-Szegö problem. Additionally, we provide bound estimates for the coefficients and an upper bound estimate for the second Hankel determinant for functions belonging to the category of analytical and bi-univalent functions. This investigation incorporates the utilization of Euler polynomials. Full article

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

15 pages, 334 KB

Open AccessArticle

Initial Coefficient Bounds Analysis for Novel Subclasses of Bi-Univalent Functions Linked with Lucas-Balancing Polynomials

by Sondekola Rudra Swamy, Daniel Breaz, Kala Venugopal, Mamatha Paduvalapattana Kempegowda, Luminita-Ioana Cotîrlă and Eleonora Rapeanu

Mathematics 2024, 12(9), 1325; https://doi.org/10.3390/math12091325 - 26 Apr 2024

Cited by 5 | Viewed by 937

Abstract

We investigate some subclasses of regular and bi-univalent functions in the open unit disk that are associated with Lucas-Balancing polynomials in this work. For functions that belong to these subclasses, we obtain upper bounds on their initial coefficients. The Fekete–Szegö problem is also [...] Read more.

We investigate some subclasses of regular and bi-univalent functions in the open unit disk that are associated with Lucas-Balancing polynomials in this work. For functions that belong to these subclasses, we obtain upper bounds on their initial coefficients. The Fekete–Szegö problem is also discussed. Along with presenting some new results, we also explore pertinent connections to earlier findings. Full article

17 pages, 332 KB

Open AccessArticle

Starlike Functions of the Miller–Ross-Type Poisson Distribution in the Janowski Domain

by Gangadharan Murugusundaramoorthy, Hatun Özlem Güney and Daniel Breaz

Mathematics 2024, 12(6), 795; https://doi.org/10.3390/math12060795 - 8 Mar 2024

Cited by 3 | Viewed by 1137

Abstract

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions

f,

utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained [...] Read more.

In this paper, considering the various important applications of Miller–Ross functions in the fields of applied sciences, we introduced a new class of analytic functions

f,

utilizing the concept of Miller–Ross functions in the region of the Janowski domain. Furthermore, we obtained initial coefficients of Taylor series expansion of

f,

coefficient inequalities for

f^{- 1}

and the Fekete–Szegö problem. We also covered some key geometric properties for functions f in this newly formed class, such as the necessary and sufficient condition, convex combination, sequential subordination and partial sum findings. Full article

11 pages, 293 KB

Open AccessArticle

Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions

by Isra Al-Shbeil, Abbas Kareem Wanas, Hala AlAqad, Adriana Cătaş and Hanan Alohali

Symmetry 2024, 16(2), 218; https://doi.org/10.3390/sym16020218 - 11 Feb 2024

Cited by 5 | Viewed by 1450

Abstract

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by

D_{Σ} (δ, η, λ, t, r)

. These functions are connected to the Bazilevič functions and the

λ

-pseudo-starlike functions. [...] Read more.

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by

D_{Σ} (δ, η, λ, t, r)

. These functions are connected to the Bazilevič functions and the

λ

-pseudo-starlike functions. We employ Sakaguchi Type Functions and Horadam polynomials in our survey. We establish the Fekete-Szegö inequality for the functions in

D_{Σ} (δ, η, λ, t, r)

and derive upper bounds for the initial Taylor–Maclaurin coefficients

| a_{2} |

and

| a_{3} |

. Additionally, we establish connections between our results and previous research papers on this topic. Full article

(This article belongs to the Special Issue Selected Papers from International Symposium on Geometric Function Theory and Applications (GFTA))

Search Results (82)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (82)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI