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17 pages, 308 KB

Open AccessArticle

Bounds for Hankel and Toeplitz Determinants of Certain Subclasses of Analytic Functions Associated with Quantum Calculus and Quasi-Subordination

by Zahid Shareef, Adriana Catas, Adeel Ahmad, Aamina Bibi, Saqib Hussain and Fethiye Müge Sakar

Axioms 2026, 15(7), 485; https://doi.org/10.3390/axioms15070485 - 29 Jun 2026

Viewed by 306

Abstract

In our present study, we introduce and examine several new subclasses of analytic functions defined through the convolution operator

H_{Y}^{λ, q}

, which is formulated using the classical error function. Our approach relies on the concept of quasi-subordination, a broad [...] Read more.

In our present study, we introduce and examine several new subclasses of analytic functions defined through the convolution operator

H_{Y}^{λ, q}

, which is formulated using the classical error function. Our approach relies on the concept of quasi-subordination, a broad extension of subordination in geometric function theory. For each of these newly defined classes, we focus on deriving significant properties such as the upper bounds of the first few Taylor–Maclaurin coefficients of normalized series, evaluation of classical Fekete–Szegö functional, and calculating the upper bounds of Hankel and Toeplitz determinants for different orders. The combination of the proposed operator and the quasi-subordination framework offers a unified strategy for tackling these problems. Our findings also generalize various existing results in the field. Full article

(This article belongs to the Special Issue Theory of Functions and Applications, 3rd Edition)

11 pages, 248 KB

Open AccessArticle

Coefficient Estimates for Bi-Univalent Functions Associated with a Third-Order Logarithmic-Type Operator

by Adnan Ghazy Alamoush, Abbas Kareem Wanas and Alina Alb Lupaş

Mathematics 2026, 14(13), 2296; https://doi.org/10.3390/math14132296 - 28 Jun 2026

Viewed by 337

Abstract

In this paper, we introduce a new class of bi-univalent functions defined by a third-order logarithmic-type differential operator. By using the subordination principle and Carathéodory functions, we investigate the coefficient estimates for the Taylor-Maclaurin coefficients

| a_{2} |

and [...] Read more.

In this paper, we introduce a new class of bi-univalent functions defined by a third-order logarithmic-type differential operator. By using the subordination principle and Carathéodory functions, we investigate the coefficient estimates for the Taylor-Maclaurin coefficients

| a_{2} |

and

| a_{3} |

. Furthermore, we derive the Fekete–Szegö inequality and obtain bounds for the second Hankel determinant

H_{2} (2)

associated with this class. Several consequences of the main results are also discussed. Full article

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

16 pages, 342 KB

Open AccessArticle

Coefficient Estimates for Analytic and Bi-Univalent Functions Classes Defined by Generalized Mathieu-Type Power Series

by Feras Yousef, Tariq Al-Hawary, Khadeejah Rasheed Alhindi and Hamed Obiedat

Mathematics 2026, 14(11), 1822; https://doi.org/10.3390/math14111822 - 24 May 2026

Viewed by 347

Abstract

In this paper, we introduce two novel subclasses,

{MS}_{κ}^{*} (δ, β, μ)

and

{NS}_{κ}^{*} (φ)

, of analytic and bi-univalent functions associated with generalized Mathieu-type power series in the open unit disk. By [...] Read more.

In this paper, we introduce two novel subclasses,

{MS}_{κ}^{*} (δ, β, μ)

and

{NS}_{κ}^{*} (φ)

, of analytic and bi-univalent functions associated with generalized Mathieu-type power series in the open unit disk. By employing coefficient-based techniques, we derive new bounds for the initial Taylor–Maclaurin coefficients and the Fekete–Szegö functional for functions belonging to these subclasses. The obtained results contribute to the ongoing development of coefficient problems in geometric function theory and provide a unified framework that extends several known results in the literature. Additionally, we present a range of special cases that recover previously studied function classes, thereby highlighting the flexibility and applicability of the proposed approach. Full article

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

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11 pages, 263 KB

Open AccessArticle

Initial Coefficient Behavior of Bi-Univalent Functions Defined Through Bernoulli Polynomial Subordination

by Mohamed Illafe, Abdulmtalb Hussen and Feras Yousef

Mathematics 2026, 14(10), 1712; https://doi.org/10.3390/math14101712 - 16 May 2026

Viewed by 308

Abstract

The study of coefficient problems for bi-univalent functions continues to play a central role in geometric function theory due to its analytical depth and wide range of applications. In this paper, we introduce a new subclass of bi-univalent functions defined through subordination to [...] Read more.

The study of coefficient problems for bi-univalent functions continues to play a central role in geometric function theory due to its analytical depth and wide range of applications. In this paper, we introduce a new subclass of bi-univalent functions defined through subordination to the generating function of Bernoulli polynomials. We derive explicit upper bounds for the initial Taylor–Maclaurin coefficients and establish a corresponding Fekete–Szegö-type inequality for functions in this class. The results obtained provide refined estimates that extend several known findings in the literature and reveal the effectiveness of Bernoulli polynomial subordination as a unifying framework for investigating coefficient problems in the theory of bi-univalent functions. Various special cases are also discussed to demonstrate the scope and applicability of the main results. Full article

(This article belongs to the Special Issue New Trends in Polynomials and Mathematical Analysis)

11 pages, 264 KB

Open AccessArticle

A Class of Bi-Bazilevič Mappings Generated via Miller-Ross Type Poisson Distribution Subordinate to Chebyshev Polynomials

by Saba N. Al-Khafaji and Emad Kadhim Mouajeeb

AppliedMath 2026, 6(5), 73; https://doi.org/10.3390/appliedmath6050073 - 7 May 2026

Viewed by 297

Abstract

Bazilevič mappings are considered very important in the theory of geometric mappings because they provide a way to generalize and study the properties of important classes of univalent mappings. Their importance is not only in the deepening of the theory, but also in [...] Read more.

Bazilevič mappings are considered very important in the theory of geometric mappings because they provide a way to generalize and study the properties of important classes of univalent mappings. Their importance is not only in the deepening of the theory, but also in the practical means of modeling phenomena in applied science and engineering, physics, and differential equations. This paper, in this sense, provides a new subclass of bi-Bazilevič mappings with the use of advanced analytical methods, Chebyshev polynomials on one side, and a Miller–Ross-type Poisson distribution on the other side. The Poisson distribution is considered one of the most important models of probability distributions with a large scope of application in the various sciences. The main components of this study are the definition and the study of this new class of functions, in which the initial Taylor–Maclaurin coefficients, in particular,

q_{2}

and

q_{3}

, are determined and estimated for mappings in this subclass. Also, the classical Fekete–Szegö problem is solved and the first-order limits of this important functional are obtained with respect to the newly introduced bi-Bazilevič mappings. The outcomes contribute to expanding both the theoretical and practical aspects of this type of mapping. Full article

(This article belongs to the Section Deterministic Mathematics)

14 pages, 533 KB

Open AccessArticle

Applications of Fractional Calculus and Quantum Calculus in Subordination and q-Derivative Operators

by Maram Alossaimi, Tseu Suet Yie, Aini Janteng and Muhammad Abbas

Fractal Fract. 2026, 10(5), 313; https://doi.org/10.3390/fractalfract10050313 - 6 May 2026

Viewed by 461

Abstract

The theory of analytic functions remains a fundamental area of geometric function theory, with particular emphasis on coefficient problems, differential subordinations, and determinant estimates. Motivated by recent developments in fractional calculus and quantum calculus, this paper introduces two new subclasses of normalized analytic [...] Read more.

The theory of analytic functions remains a fundamental area of geometric function theory, with particular emphasis on coefficient problems, differential subordinations, and determinant estimates. Motivated by recent developments in fractional calculus and quantum calculus, this paper introduces two new subclasses of normalized analytic functions by employing the subordination principle in combination with the q-derivative operator and the q-Sălăgeăn differential operator within the framework of quantum calculus. The inclusion of fractional and q-calculus techniques provides a more flexible and generalized approach to classical problems in complex analysis, enabling deeper structural insights into analytic function classes. Using the subordination framework, we derive coefficient relations for the proposed subclasses. Furthermore, we establish sharp upper bounds for the Fekete–Szegö functional

| a_{3} - δ a_{2}^{2} |

and for the second Hankel determinant

H_{2, 2} (f) = a_{2} a_{4} - a_{3}^{2}

. The obtained results extend and unify several known works in the literature and demonstrate how the interaction between fractional calculus, quantum operators, and subordination theory can be effectively used in geometric function theory. Finally, the presented approach opens the door for further investigations involving higher-order Hankel determinants, other subclasses of analytic functions, and potential extensions involving special functions and fractional operators. Full article

(This article belongs to the Special Issue Fractional Calculus, Quantum Calculus and Special Functions in Complex Analysis)

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25 pages, 374 KB

Open AccessArticle

Some New Subclasses of Bi-Univalent Functions Related to Quantum Calculus

by Renjie Guo, Sadia Riaz, Wajiha Bushra, Adeel Ahmad, Saqib Hussain and Saima Noor

Mathematics 2026, 14(5), 911; https://doi.org/10.3390/math14050911 - 7 Mar 2026

Viewed by 517

Abstract

The primary objective of this paper is to introduce and investigate several novel subclasses of bi-univalent functions associated with the q-calculus framework. Using appropriate analytical techniques, we derive coefficient bounds for the initial coefficients of the functions belonging to these newly defined [...] Read more.

The primary objective of this paper is to introduce and investigate several novel subclasses of bi-univalent functions associated with the q-calculus framework. Using appropriate analytical techniques, we derive coefficient bounds for the initial coefficients of the functions belonging to these newly defined classes. In particular, we provide explicit estimates for the second-order Hankel determinant and address the classical Fekete–Szegö functional problem within the context of these classes under suitable conditions. It is important to note that the findings presented in this work not only contribute to the ongoing development of q-analogs in geometric function theory, but also serve as a unifying generalization of many previously known results, which are obtained as special cases of our main findings. Full article

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

14 pages, 330 KB

Open AccessArticle

Comprehensive Subfamilies of Bi-Univalent Functions Involving a Certain Operator Subordinate to Generalized Bivariate Fibonacci Polynomials

by Ibtisam Aldawish, Hari M. Srivastava, Sheza M. El-Deeb and Tamer M. Seoudy

Mathematics 2026, 14(2), 292; https://doi.org/10.3390/math14020292 - 13 Jan 2026

Cited by 1 | Viewed by 651

Abstract

This paper introduces novel subfamilies of analytic and bi-univalent functions in

Ω = \{ς \in C : | ς | < 1\}

, defined by applying a linear operator associated with the Mittag–Leffler function and requiring subordination to domains related to generalized bivariate [...] Read more.

This paper introduces novel subfamilies of analytic and bi-univalent functions in

Ω = \{ς \in C : | ς | < 1\}

, defined by applying a linear operator associated with the Mittag–Leffler function and requiring subordination to domains related to generalized bivariate Fibonacci polynomials. The proposed framework provides a unified treatment that generalizes numerous earlier studies by incorporating parameters controlling both the operator’s fractional calculus features and the domain’s combinatorial geometry. For these subfamilies, we establish initial coefficient bounds (

|d_{2}|

,

|d_{3}|

) and solve the Fekete–Szegö problem (

|d_{3} - ξ d_{2}^{2}|

). The derived inequalities are interesting, and their proofs leverage the intricate interplay between the series expansions of the Mittag–Leffler function and the generating function of the Fibonacci polynomials. By specializing the parameters governing the operator and the polynomial domain, we show how our main theorems systematically recover and extend a wide range of known results from the literature, thereby demonstrating the generality and unifying power of our approach. Full article

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

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

Cited by 1 | Viewed by 781

Abstract

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 1058

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

Cited by 1 | Viewed by 966

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)

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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 3 | Viewed by 1910

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, 265 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 3 | Viewed by 1280

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 6 | Viewed by 1630

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 1943

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)

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