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15 pages, 845 KB

Open AccessArticle

Third-Order Hankel Determinant for a Class of Bi-Univalent Functions Associated with Sine Function

by Mohammad El-Ityan, Mustafa A. Sabri, Suha Hammad, Basem Frasin, Tariq Al-Hawary and Feras Yousef

Mathematics 2025, 13(17), 2887; https://doi.org/10.3390/math13172887 (registering DOI) - 6 Sep 2025

Abstract

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to

1 +

sinz. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus [...] Read more.

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to

1 +

sinz. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function

\sqrt{1 + tanh z}

, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems. Full article

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

19 pages, 2680 KB

Open AccessArticle

Sharp Bounds and Electromagnetic Field Applications for a Class of Meromorphic Functions Introduced by a New Operator

by Abdelrahman M. Yehia, Atef F. Hashem, Samar M. Madian and Mohammed M. Tharwat

Axioms 2025, 14(9), 684; https://doi.org/10.3390/axioms14090684 (registering DOI) - 5 Sep 2025

Abstract

In this paper, we present a new integral operator that acts on a class of meromorphic functions on the punctured unit disc

U^{*}

. This operator enables the definition of a new subclass of meromorphic univalent functions. We obtain sharp bounds for [...] Read more.

In this paper, we present a new integral operator that acts on a class of meromorphic functions on the punctured unit disc

U^{*}

. This operator enables the definition of a new subclass of meromorphic univalent functions. We obtain sharp bounds for the Fekete–Szegö inequality and the second Hankel determinant for this class. The theoretical approach is based on differential subordination. Furthermore, we link these theoretical insights to applications in 2D electromagnetic field theory by outlining a physical framework in which the operator functions as a field transformation kernel. We show that the operator’s parameters correspond to physical analogs of field regularization and spectral redistribution, and we use subordination theory to simulate the design of vortex-free fields. The findings provide new insights into the interaction between geometric function theory and physical field modeling. Full article

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

23 pages, 345 KB

Open AccessFeature PaperArticle

On Certain Subclasses of Analytic Functions Associated with a Symmetric q-Differential Operator

by Vasile-Aurel Caus

Mathematics 2025, 13(17), 2860; https://doi.org/10.3390/math13172860 - 4 Sep 2025

Abstract

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities [...] Read more.

This paper explores a class of analytic functions defined in the open unit disk by means of a symmetric q-differential operator. In the first part, we derive sufficient conditions for functions to belong to a subclass associated with this operator, using inequalities involving their coefficients. Additionally, we establish several inclusion relations between these subclasses, obtained by varying the defining parameters. In the second part, we focus on differential subordination and superordination for functions transformed by the operator. We provide sufficient conditions under which such functions are subordinate or superordinate to univalent functions, and we determine the best dominant and best subordinant in specific cases. These results are complemented by several corollaries that highlight particular instances of the main theorems. Furthermore, we present a sandwich-type result that brings together the subordination and superordination frameworks in a unified analytic statement. Full article

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

18 pages, 3211 KB

Open AccessArticle

Sharp Results and Fluid Flow Applications for a Specific Class of Meromorphic Functions Introduced by a New Operator

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

Axioms 2025, 14(8), 620; https://doi.org/10.3390/axioms14080620 - 8 Aug 2025

Viewed by 301

Abstract

In this investigation, we introduce a new meromorphic operator defined by meromorphic univalent functions. A new class of meromorphic functions is introduced by this operator, which can generate several distinct subclasses depending on the values of its parameters. Within the framework of this [...] Read more.

In this investigation, we introduce a new meromorphic operator defined by meromorphic univalent functions. A new class of meromorphic functions is introduced by this operator, which can generate several distinct subclasses depending on the values of its parameters. Within the framework of this class of functions, we obtain several significant algebraic and geometric properties, including coefficient estimates, distortion theorems, the radius of starlikeness, convex combination closure, extreme point characterization, and neighborhood structure. Our findings are sharp, offering accurate and significant insights into the mathematical structure and behavior of these functions. In addition, we present several applications of these results in fluid mechanics, like identifying stagnation points in vortex flows, predicting velocity decline in source/sink systems, and determining stability thresholds that protect crucial streamlines from perturbations, which demonstrates that the introduced operator and class characterize critical properties of 2D potential flows. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

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13 pages, 345 KB

Open AccessArticle

An Application of Liouville–Caputo-Type Fractional Derivatives on Certain Subclasses of Bi-Univalent Functions

by Ibtisam Aldawish, Hari M. Srivastava, Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy and Kaliappan Vijaya

Fractal Fract. 2025, 9(8), 505; https://doi.org/10.3390/fractalfract9080505 - 31 Jul 2025

Viewed by 341

Abstract

In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients

| c_{2} |

,

| c_{3} |

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

In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients

| c_{2} |

,

| c_{3} |

for functions in these subclasses of bi-univalent functions. Additionally, by using the values of

a_{2}, a_{3}

we determine the Fekete–Szegö inequality results. Moreover, a few new subclasses are deduced that have not been studied in relation to Liouville–Caputo fractional derivatives so far. The implications of the results are also emphasized. Our results are concrete examples of several earlier discoveries that are not only improved but also expanded upon. Full article

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

21 pages, 352 KB

Open AccessArticle

Inverse and Logarithmic Coefficient Bounds of Concave Univalent Functions

by Kuppusami Sakthivel, Nak Eun Cho and Srikandan Sivasubramanian

Axioms 2025, 14(8), 553; https://doi.org/10.3390/axioms14080553 - 22 Jul 2025

Viewed by 353

Abstract

The concept of coefficient estimates on univalent functions is one of the interesting aspects explored recently by many researchers. Motivated by this direction, in this present work, we obtain the upper bounds of initial inverse coefficients and logarithmic coefficients and the upper bounds [...] Read more.

The concept of coefficient estimates on univalent functions is one of the interesting aspects explored recently by many researchers. Motivated by this direction, in this present work, we obtain the upper bounds of initial inverse coefficients and logarithmic coefficients and the upper bounds of differences between these successive coefficients related to concave univalent functions. Further, we also calculate the upper bounds of third-order Hankel, Toeplitz, and Vandermonde determinants in terms of specified coefficients connected to concave univalent functions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications IV)

18 pages, 608 KB

Open AccessArticle

The Geometric Characterizations of the Ramanujan-Type Entire Function

by Khaled Mehrez and Abdulaziz Alenazi

Mathematics 2025, 13(14), 2301; https://doi.org/10.3390/math13142301 - 18 Jul 2025

Viewed by 304

Abstract

In the present paper, we present certain geometric properties, such as starlikeness, convexity of order

η (0 \leq η < 1)

, and close-to-convexity, in an open unit disk of the normalized form of Ramanujan-type entire functions. As a consequence, a [...] Read more.

In the present paper, we present certain geometric properties, such as starlikeness, convexity of order

η (0 \leq η < 1)

, and close-to-convexity, in an open unit disk of the normalized form of Ramanujan-type entire functions. As a consequence, a specific range of parameters is derived such that this function belongs to Hardy spaces

H^{\infty}

and

H^{r} .

Finally, as an application, we present the monotonicity property of the Ramanujan-type entire function using the method of subordination factor sequences. Full article

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

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16 pages, 291 KB

Open AccessArticle

Initial Coefficient Bounds for Bi-Close-to-Convex and Bi-Quasi-Convex Functions with Bounded Boundary Rotation Associated with q-Sălăgean Operator

by Prathviraj Sharma, Srikandan Sivasubramanian, Adriana Catas and Sheza M. El-Deeb

Mathematics 2025, 13(14), 2252; https://doi.org/10.3390/math13142252 - 11 Jul 2025

Viewed by 560

Abstract

In this article, through the application of the q-Sălăgean operator associated with functions characterized by bounded boundary rotation, we propose a few new subclasses of bi-univalent functions that utilize the q-Sălăgean operator with bounded boundary rotation in the open unit disk [...] Read more.

In this article, through the application of the q-Sălăgean operator associated with functions characterized by bounded boundary rotation, we propose a few new subclasses of bi-univalent functions that utilize the q-Sălăgean operator with bounded boundary rotation in the open unit disk

E

. For these classes, we establish the initial bounds for the coefficients

| a_{2} |

and

| a_{3} |

. Additionally, we have derived the well-known Fekete–Szegö inequality for this newly defined subclasses. Full article

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

19 pages, 296 KB

Open AccessArticle

Applications of q-Bessel-Struve Functions on Univalent Functions

by Saddaf Noreen, Saiful R. Mondal, Muhey U. Din, Saima Mushtaq, Zhang Wei and Adil Murtaza

Mathematics 2025, 13(13), 2150; https://doi.org/10.3390/math13132150 - 30 Jun 2025

Viewed by 256

Abstract

In this paper, the authors derived some new sufficient conditions for q-close-to-convexity with respect to certain functions involving three different normalizations of q-Bessel–Struve functions. These new inequalities, under which the three normalizations of q-Bessel–Struve functions are q-close-to-convex associated with [...] Read more.

In this paper, the authors derived some new sufficient conditions for q-close-to-convexity with respect to certain functions involving three different normalizations of q-Bessel–Struve functions. These new inequalities, under which the three normalizations of q-Bessel–Struve functions are q-close-to-convex associated with certain functions, hold for

v \geq \frac{- 3}{2}

and for all

q \in (0, 1)

. The work is new and has great importance because it shows the pivotal role between the q-special functions and geometric function theory. Full article

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

14 pages, 569 KB

Open AccessArticle

A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator

by Mohammad El-Ityan, Tariq Al-Hawary, Basem Aref Frasin and Ibtisam Aldawish

Symmetry 2025, 17(7), 982; https://doi.org/10.3390/sym17070982 - 21 Jun 2025

Cited by 2 | Viewed by 554

Abstract

In this work, we introduce a new subclass of bi-univalent functions using the

(p, q)

-derivative operator and the concept of subordination to generalized Laguerre polynomials

L_{t}^{ς} (k)

, which satisfy the differential equation [...] Read more.

In this work, we introduce a new subclass of bi-univalent functions using the

(p, q)

-derivative operator and the concept of subordination to generalized Laguerre polynomials

L_{t}^{ς} (k)

, which satisfy the differential equation

k y^{″} + (1 + ς - k) y^{'} + t y = 0,

with

1 + ς > 0

,

k \in R

, and

t \geq 0

. We focus on functions that blend the geometric features of starlike and convex mappings in a symmetric setting. The main goal is to estimate the initial coefficients of functions in this new class. Specifically, we obtain sharp upper bounds for

| a_{2} |

and

| a_{3} |

and for the Fekete–Szegö functional

| a_{3} - η a_{2}^{2} |

for some real number

η

. In the final section, we explore several special cases that arise from our general results. These results contribute to the ongoing development of bi-univalent function theory in the context of

(p, q)

-calculus. Full article

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

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9 pages, 265 KB

Open AccessArticle

Sufficient and Necessary Conditions for Generalized Distribution Series on Comprehensive Subclass of Analytic Functions

by Tariq Al-Hawary, Basem Frasin and Ibtisam Aldawish

Mathematics 2025, 13(12), 2029; https://doi.org/10.3390/math13122029 - 19 Jun 2025

Viewed by 491

Abstract

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series [...] Read more.

In this paper, we demonstrate a relationship between a generalized distribution series and a comprehensive subclass of analytic functions. The primary aim of this study is to determine a necessary and sufficient condition for the generalized distribution series

E_{ϕ}^{*} (ς, z)

to belong to the inclusive subclass

Π_{η} (Q_{3}, Q_{2}, Q_{1}, Q_{0})

. Necessary and sufficient conditions are also given for the generalized distribution series

E_{ϕ}^{*} (ς, z) ℏ

and the integral operator

J_{ς}^{ϕ} (z)

to be in the inclusive subclass

Π_{η} (Q_{3}, Q_{2}, Q_{1}, 0)

. Further, we provide a number of corollaries, which improve the existing ones that are available in some recent studies. The results presented here not only improve the earlier studies, but also give rise to a number of new results for particular choices of

Q_{3}, Q_{2}, Q_{1}

and

Q_{0}

. Full article

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

13 pages, 762 KB

Open AccessArticle

Starlike Functions with Respect to (ℓ, κ)-Symmetric Points Associated with the Vertical Domain

by Daniel Breaz, Kadhavoor R. Karthikeyan and Dharmaraj Mohankumar

Symmetry 2025, 17(6), 933; https://doi.org/10.3390/sym17060933 - 12 Jun 2025

Viewed by 298

Abstract

The study of various subclasses of univalent functions involving the solutions to various differential equations is not totally new, but studies of analytic functions with respect to

(ℓ, κ)

-symmetric points are rarely conducted. Here, using a differential operator which [...] Read more.

The study of various subclasses of univalent functions involving the solutions to various differential equations is not totally new, but studies of analytic functions with respect to

(ℓ, κ)

-symmetric points are rarely conducted. Here, using a differential operator which was defined using the Hadamard product of Mittag–Leffler function and general analytic function, we introduce a new class of starlike functions with respect to

(ℓ, κ)

-symmetric points associated with the vertical domain. To define the function class, we use a Carathéodory function which was recently introduced to study the impact of various conic regions on the vertical domain. We obtain several results concerned with integral representations and coefficient inequalities for functions belonging to this class. The results obtained by us here not only unify the recent studies associated with the vertical domain but also provide essential improvements of the corresponding results. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Applications, 2nd Edition)

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16 pages, 613 KB

Open AccessArticle

A Study of Certain Geometric Properties and Hardy Spaces of the Normalized Miller-Ross Function

by Muhammad Abubakr, Mohsan Raza, Abdulaziz Alenazi and Khaled Mehrez

Mathematics 2025, 13(12), 1919; https://doi.org/10.3390/math13121919 - 8 Jun 2025

Viewed by 400

Abstract

The main objective of this research is to investigate specific sufficiency criteria for the strongly starlikeness, strongly convexity, starlikeness, convexity and pre-starlikeness of the normalized Miller-Ross function. Furthermore, we establish sufficient conditions under which the normalized Miller-Ross function belongs to Hardy spaces and [...] Read more.

The main objective of this research is to investigate specific sufficiency criteria for the strongly starlikeness, strongly convexity, starlikeness, convexity and pre-starlikeness of the normalized Miller-Ross function. Furthermore, we establish sufficient conditions under which the normalized Miller-Ross function belongs to Hardy spaces and the class-bounded analytic functions. Some of the various results which are derived in this paper are presumably new and their significance is illustrated through several interesting examples. Full article

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

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22 pages, 810 KB

Open AccessArticle

Gregory Polynomials Within Sakaguchi-Type Function Classes: Analytical Estimates and Geometric Behavior

by Arzu Akgül and Georgia Irina Oros

Symmetry 2025, 17(6), 884; https://doi.org/10.3390/sym17060884 - 5 Jun 2025

Viewed by 423

Abstract

This work introduces a novel family of analytic and univalent functions formulated through the integration of Gregory coefficients and Sakaguchi-type functions. Employing subordination techniques, we obtain sharp bounds for the initial coefficients in their Taylor expansions. The influence of parameter variations is examined [...] Read more.

This work introduces a novel family of analytic and univalent functions formulated through the integration of Gregory coefficients and Sakaguchi-type functions. Employing subordination techniques, we obtain sharp bounds for the initial coefficients in their Taylor expansions. The influence of parameter variations is examined through comprehensive geometric visualizations, which confirm the non-emptiness of the class and provide insights into its structural properties. Furthermore, Fekete–Szegö inequalities are established, enriching the theory of bi-univalent functions. The combination of analytical methods and geometric representations offers a versatile framework for future research in geometric function theory. Full article

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

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11 pages, 265 KB

Open AccessArticle

On Certain Bounds of Harmonic Univalent Functions

by Fethiye Müge Sakar, Omendra Mishra, Georgia Irina Oros and Basem Aref Frasin

Axioms 2025, 14(6), 393; https://doi.org/10.3390/axioms14060393 - 22 May 2025

Viewed by 472

Abstract

Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk

U = \{z \in C : |z| < 1\}

can be written as [...] Read more.

Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk

U = \{z \in C : |z| < 1\}

can be written as a sum

f = h + \bar{g}

, where h and g are analytic functions in

U

and are called the analytic part and the co-analytic part of f, respectively. In this paper, the harmonic shear

f = h + \bar{g} \in S_{H}

and its rotation

f^{μ}

by

μ (μ \in C, |μ| = 1)

are considered. Bounds are established for this rotation

f^{μ}

, specific inequalities that define the Jacobian of

f^{μ}

are obtained, and the integral representation is determined. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Applications, 2nd Edition)

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