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18 pages, 303 KB

Open AccessArticle

Symmetric Properties of Janowski-Type q-Harmonic Close-to-Convex Functions

by Yusra Taj, Sarfraz Nawaz Malik and Alina Alb Lupaş

Symmetry 2026, 18(5), 702; https://doi.org/10.3390/sym18050702 - 22 Apr 2026

Viewed by 234

We introduce and study a new subclass of Janowski-type harmonic close-to-convex functions in the open unit disk defined via the Jackson q-derivative operator. The structure of the operator naturally reflects certain symmetric properties in the analytic representation of the considered harmonic mappings. [...] Read more.

We introduce and study a new subclass of Janowski-type harmonic close-to-convex functions in the open unit disk defined via the Jackson q-derivative operator. The structure of the operator naturally reflects certain symmetric properties in the analytic representation of the considered harmonic mappings. By applying subordination techniques, we establish sufficient conditions for sense-preserving close-to-convexity and distortion estimates. The extreme points of the class are determined, and its topological properties are examined, showing that the class is convex and compact. We further obtain the radius of starlikeness and prove that the class is closed under convolution. Moreover, as

q \to 1^{-}

, the operator reduces to the classical derivative, and our results recover several known results in the existing literature, demonstrating that the present work extends and generalizes earlier findings. Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

17 pages, 749 KB

Open AccessArticle

Further Geometric Behavior of the Generalized Marcum Q-Function

by Khaled Mehrez, Abdulaziz Alenazi and Mohsan Raza

Symmetry 2026, 18(3), 467; https://doi.org/10.3390/sym18030467 - 9 Mar 2026

Viewed by 312

Abstract

In this paper, we investigate a class of analytic functions associated with the generalized Marcum Q-function and its Alexander transform. We establish sufficient conditions under which these functions exhibit important geometric properties in the open unit disk, including strong starlikeness, strong convexity, [...] Read more.

In this paper, we investigate a class of analytic functions associated with the generalized Marcum Q-function and its Alexander transform. We establish sufficient conditions under which these functions exhibit important geometric properties in the open unit disk, including strong starlikeness, strong convexity, and pre-starlikeness. The results presented are believed to be new and are supported by illustrative examples and consequences. Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

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9 pages, 241 KB

Open AccessFeature PaperArticle

Some Remarks on Ozaki, Ono and Umezawa’s Results

by Mamoru Nunokawa, Hitoshi Saitoh, Janusz Sokół and Edyta Trybucka

Mathematics 2026, 14(5), 870; https://doi.org/10.3390/math14050870 - 4 Mar 2026

Viewed by 337

Abstract

Recall that in On a general second order derivative, Sci. Rep. Tokyo Kyoiku Daigaku A, 5(124–127)(1956), 111–114, Ozaki, Ono and Umezawa proved a result that if

f (z)

is analytic and satisfies [...] Read more.

Recall that in On a general second order derivative, Sci. Rep. Tokyo Kyoiku Daigaku A, 5(124–127)(1956), 111–114, Ozaki, Ono and Umezawa proved a result that if

f (z)

is analytic and satisfies

| f^{″} (z) | < 1

in the unit disc

D = \{z : |z| < 1\}

, then

| f^{'} (z) - 1 | < 1

and so,

f (z)

is univalent in

D

, because

Re {f^{'} (z)} > 0

in

D

implies univalence by the Noshiro–Warschawski Theorem. In this paper, we obtain another sufficient condition for univalence of

f (z)

by applying a hypothesis for modulus of

arg {f^{″} (z)}

. Full article

(This article belongs to the Section C: Mathematical Analysis)

21 pages, 730 KB

Open AccessArticle

Certain Geometric Properties of Normalized Euler Polynomial

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

Fractal Fract. 2026, 10(3), 136; https://doi.org/10.3390/fractalfract10030136 - 24 Feb 2026

Viewed by 461

Abstract

In this paper, we introduce and investigate a new class of analytic functions generated by Euler polynomials through a suitable normalization. Using classical tools from geometric function theory, including coefficient monotonicity, Fejér-type inequalities, MacGregor’s criteria, and Ozaki’s close-to-convexity condition, we establish sufficient conditions [...] Read more.

In this paper, we introduce and investigate a new class of analytic functions generated by Euler polynomials through a suitable normalization. Using classical tools from geometric function theory, including coefficient monotonicity, Fejér-type inequalities, MacGregor’s criteria, and Ozaki’s close-to-convexity condition, we establish sufficient conditions for the univalence, starlikeness, convexity, and close-to-convexity of the proposed Euler-polynomial-based normalized function. Sharp radius results for starlikeness, convexity, and close-to-convexity in the disk

D_{1 / 2}

are derived by exploiting refined coefficient bounds involving higher-order Euler polynomial terms. Several illustrative examples and graphical demonstrations are provided to verify the theoretical findings. The results obtained extend the known geometric properties of special function-based analytic classes and offer a new perspective on the geometric behavior of Euler polynomials in the unit disk. Full article

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

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17 pages, 357 KB

Open AccessArticle

Novel Bi-Univalent Subclasses Generated by the q-Analogue of the Ruscheweyh Operator and Hermite Polynomials

by Feras Yousef, Tariq Al-Hawary, Mohammad El-Ityan and Ibtisam Aldawish

Mathematics 2026, 14(2), 382; https://doi.org/10.3390/math14020382 - 22 Jan 2026

Cited by 2 | Viewed by 487

Abstract

This work introduces new bi-univalent function classes defined using the fractional q-Ruscheweyh operator and characterized by subordination to q-Hermite polynomials. We derive coefficient bounds and Fekete–Szegö inequalities for these classes and show that our results generalize several earlier findings in both [...] Read more.

This work introduces new bi-univalent function classes defined using the fractional q-Ruscheweyh operator and characterized by subordination to q-Hermite polynomials. We derive coefficient bounds and Fekete–Szegö inequalities for these classes and show that our results generalize several earlier findings in both the classical and q-analytic settings. The approach highlights the effectiveness of q-Hermite structures in analyzing operator-defined subclasses of bi-univalent functions. Full article

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

17 pages, 843 KB

Open AccessArticle

Lemniscate Starlikeness and Convexity for the Generalized Marcum Q-Function

by Khaled Mehrez and Abdulaziz Alenazi

Mathematics 2026, 14(2), 364; https://doi.org/10.3390/math14020364 - 21 Jan 2026

Viewed by 368

Abstract

In this paper, we investigate new geometric properties of normalized analytic functions associated with the generalized Marcum Q-function. In particular, we focus on two analytic forms derived from a normalized derivative of a representation involving the Marcum Q-function, and its Alexander [...] Read more.

In this paper, we investigate new geometric properties of normalized analytic functions associated with the generalized Marcum Q-function. In particular, we focus on two analytic forms derived from a normalized derivative of a representation involving the Marcum Q-function, and its Alexander transform. For these functions, we establish sufficient conditions ensuring membership in the classes of lemniscate starlike and lemniscate convex functions. Special attention is given to the case

ν = 1

, where explicit admissible parameter ranges for b are derived. We further examine inclusion relations between these normalized analytic forms and lemniscate subclasses, complemented by several corollaries, illustrative examples, and graphical visualizations. These results extend and enrich the geometric function theory of special functions related to the generalized Marcum Q-function. Full article

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

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13 pages, 534 KB

Open AccessArticle

Certain Geometric Investigations of Three Normalized Bessel-Type Functions of a Complex Variable

by Rabab Alyusof, Shams Alyusof, Rabha M. El-Ashwah and Alaa H. El-Qadeem

Mathematics 2025, 13(23), 3888; https://doi.org/10.3390/math13233888 - 4 Dec 2025

Viewed by 433

Abstract

We recall the normalized forms for the three Bessel-type functions; these functions are the Bessel function, Lommel function, and Struve function of the first kind. By using convolution, we define normalized forms. The essential purpose is to introduce necessary and sufficient bounds of [...] Read more.

We recall the normalized forms for the three Bessel-type functions; these functions are the Bessel function, Lommel function, and Struve function of the first kind. By using convolution, we define normalized forms. The essential purpose is to introduce necessary and sufficient bounds of these normalized functions so these functions are starlike and convex of order

γ

and type

δ

. Full article

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

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16 pages, 1247 KB

Open AccessArticle

Sharp Coefficient Bounds for a Class of Analytic Functions Related to Exponential Function

by Adel Salim Tayyah, Sibel Yalçın and Hasan Bayram

Mathematics 2025, 13(23), 3878; https://doi.org/10.3390/math13233878 - 3 Dec 2025

Cited by 2 | Viewed by 696

Abstract

In this paper, we introduce a new class of analytic functions, denoted by

S (ν, φ_{ϑ, e})

, and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter [...] Read more.

In this paper, we introduce a new class of analytic functions, denoted by

S (ν, φ_{ϑ, e})

, and provide illustrative examples to elucidate its properties. This class generalizes the starlike and convex functions previously defined by Khatter et al. in relation to the exponential function. A significant contribution of this work is the derivation of sharp bounds for various coefficient-related problems within this class. The computational challenges involved in deriving these bounds were effectively addressed using Mathematica^TM codes. Additionally, figures illustrating the geometric properties and essential computations have been incorporated into the paper. Full article

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

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16 pages, 282 KB

Open AccessArticle

On a Unified Subclass of Analytic Functions with Negative Coefficients Defined via a Generalized q-Calculus Operator

by Mohamed Illafe and Feras Yousef

AppliedMath 2025, 5(4), 158; https://doi.org/10.3390/appliedmath5040158 - 7 Nov 2025

Cited by 1 | Viewed by 574

Abstract

We introduce and analyze a subclass of analytic functions with negative coefficients, denoted by

P_{q, σ}^{m, ℓ, p} (α, η)

, constructed through a generalized q-calculus operator in combination with a multiplier-type transformation. For [...] Read more.

We introduce and analyze a subclass of analytic functions with negative coefficients, denoted by

P_{q, σ}^{m, ℓ, p} (α, η)

, constructed through a generalized q-calculus operator in combination with a multiplier-type transformation. For this class, we obtain sharp coefficient bounds, growth and distortion estimates, and closure results. The radii of close-to-convexity, starlikeness, and convexity are determined, and further consequences, such as integral means inequalities and neighborhood characterizations, are derived. The results presented provide a broad framework that incorporates and extends several earlier families of analytic and geometric function classes. Full article

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17 pages, 307 KB

Open AccessArticle

Generalization of the Rafid Operator and Its Symmetric Role in Meromorphic Function Theory with Electrostatic Applications

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

Symmetry 2025, 17(11), 1837; https://doi.org/10.3390/sym17111837 - 2 Nov 2025

Viewed by 468

Abstract

This study introduces a new integral operator

I_{p, μ}^{δ}

that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses:

Σ_{p}^{+} (δ, μ, α)

, consisting [...] Read more.

This study introduces a new integral operator

I_{p, μ}^{δ}

that extends the traditional Rafid operator to meromorphic p-valent functions. Using this operator, we define and investigate two new subclasses:

Σ_{p}^{+} (δ, μ, α)

, consisting of functions with nonnegative coefficients, and

Σ_{p}^{+} (δ, μ, α, c)

, which further fixes the second positive coefficient. For these classes, we establish a necessary and sufficient coefficient condition, which serves as the foundation for deriving a set of sharp results. These include accurate coefficient bounds, distortion theorems for functions and derivatives, and radii of starlikeness and convexity of a specific order. Furthermore, we demonstrate the closure property of the class

Σ_{p}^{+} (δ, μ, α, c)

, identify its extreme points, and then construct a neighborhood theorem. All the findings presented in this paper are sharp. To demonstrate the practical utility of our symmetric operator paradigm, we apply it to a canonical fractional electrodynamics problem. We demonstrate how sharp distortion theorems establish rigorous, time-invariant upper bounds for a solitary electrostatic potential and its accompanying electric field, resulting in a mathematically guaranteed safety buffer against dielectric breakdown. This study develops a symmetric and consistent approach to investigating the geometric characteristics of meromorphic multivalent functions and their applications in physical models. Full article

(This article belongs to the Special Issue Symmetry in Complex Analysis Operators Theory)

20 pages, 840 KB

Open AccessArticle

Sharp Functional Inequalities for Starlike and Convex Functions Defined via a Single-Lobed Elliptic Domain

by Adel Salim Tayyah, Sarem H. Hadi, Abdullah Alatawi, Muhammad Abbas and Ovidiu Bagdasar

Mathematics 2025, 13(21), 3367; https://doi.org/10.3390/math13213367 - 22 Oct 2025

Cited by 5 | Viewed by 825

Abstract

In this paper, we introduce two novel subclasses of analytic functions, namely, starlike and convex functions of Ma–Minda-type, associated with a newly proposed domain. We set sharp bounds on the basic coefficients of these classes and provide sharp estimates of the second- and [...] Read more.

In this paper, we introduce two novel subclasses of analytic functions, namely, starlike and convex functions of Ma–Minda-type, associated with a newly proposed domain. We set sharp bounds on the basic coefficients of these classes and provide sharp estimates of the second- and third-order Hankel determinants, demonstrating the power of our analytic approach, the clarity of its results, and its applicability even in unconventional domains. Full article

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

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25 pages, 393 KB

Open AccessArticle

Geometric Attributes of Analytic Functions Generated by Mittag-Leffler Function

by Ekram E. Ali, Rabha M. El-Ashwah, Wafaa Y. Kota and Abeer M. Albalahi

Mathematics 2025, 13(20), 3284; https://doi.org/10.3390/math13203284 - 14 Oct 2025

Cited by 4 | Viewed by 569

Abstract

This study examines the necessary requirements for some analytic function subclasses, especially those associated with the generalized Mittag-Leffler function, to be classified as univalent function subclasses that are determined by particular geometric constraints. The core methodology revolves around the application of the Hadamard [...] Read more.

This study examines the necessary requirements for some analytic function subclasses, especially those associated with the generalized Mittag-Leffler function, to be classified as univalent function subclasses that are determined by particular geometric constraints. The core methodology revolves around the application of the Hadamard (or convolution) product involving a normalized Mittag-Leffler function

M_{κ, χ} (ζ)

, leading to the definition of a new linear operator

S_{χ, ϑ}^{κ} ℏ (ζ) .

We investigate inclusion results in the recently defined subclasses

\tilde{Ξ} (ϖ, ϱ), \hat{L} (ϖ, ϱ), \hat{K} (ϖ, ϱ)

and

\hat{F} (ϖ, ϱ)

, which generalize the classical classes of starlike, convex, and close-to-convex functions. This is achieved by utilizing recent developments in the theory of univalent functions. In addition, we examine the behavior of functions from the class

R_{θ} (E, V)

under the action of the convolution operator

W_{χ, ϑ}^{κ} h (ζ)

, establishing sufficient criteria for the resulting images to lie within the subclasses of analytic function. Also, certain mapping properties related to these subclasses are analyzed. In addition, the geometric features of an integral operator connected to the Mittag-Leffler function are examined. A few particular cases of our main findings are also mentioned and examined and the paper ends with the conclusions regarding the obtained results. Full article

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

13 pages, 286 KB

Open AccessArticle

Categories of Harmonic Functions in the Symmetric Unit Disk Linked to the Bessel Function

by Naci Taşar, Fethiye Müge Sakar, Basem Frasin and Ibtisam Aldawish

Symmetry 2025, 17(9), 1581; https://doi.org/10.3390/sym17091581 - 22 Sep 2025

Cited by 2 | Viewed by 746

Abstract

Here in this paper, we establish the basic inclusion relations among the harmonic class

HF (σ, η)

with the classes

S_{HF}^{*}

of starlike harmonic functions and

K_{HF}

of convex harmonic functions defined in open symmetric unit disk [...] Read more.

Here in this paper, we establish the basic inclusion relations among the harmonic class

HF (σ, η)

with the classes

S_{HF}^{*}

of starlike harmonic functions and

K_{HF}

of convex harmonic functions defined in open symmetric unit disk U. Moreover, we investigate inclusion connections for the harmonic classes

T N_{HF} (ϱ)

and

T Q_{HF} (ϱ)

of harmonic functions by applying the operator

Λ

associated with the Bessel function. Furthermore, several special cases of the main results are obtained for the particular case

σ = 0

. Full article

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

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

Cited by 1 | Viewed by 799

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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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 946

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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