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21 pages, 1881 KB

Open AccessArticle

Applications of the Generalized Marcum Q-Function to Janowski Subclasses of Harmonic Functions

by Mohammad Faisal Khan and Mohammed AbaOud

Fractal Fract. 2026, 10(3), 209; https://doi.org/10.3390/fractalfract10030209 - 23 Mar 2026

Viewed by 349

In this work, we provide a convolution type operator

Λ_{ν, b}

that is produced by the generalized Marcum Q-function and examine how it maps to various Janowski-type subclasses of harmonic univalent functions. Since the Marcum Q-function has an integral [...] Read more.

In this work, we provide a convolution type operator

Λ_{ν, b}

that is produced by the generalized Marcum Q-function and examine how it maps to various Janowski-type subclasses of harmonic univalent functions. Since the Marcum Q-function has an integral form via the lower incomplete gamma function, the convolution operator

Λ_{ν, b}

can be understood as a fractional type integral operator operating on the coefficients of harmonic mappings. Applying

Λ_{ν, b}

to harmonic mappings

f = h + \bar{g}

in the unit disk

D

, we derive coefficient inequalities, and inclusion relations for various subclasses of harmonic and analytic univalent functions. In particular, we give sufficient conditions for

Λ_{ν, b} (f)

to belong to Janowski-starlike families such as

S_{H}^{*} (F, G)

,

K_{H}^{0}

, and

R_{H} (F, G)

. Closure properties of the Janowski class under the proposed operator are also established. Numerical tables and examples confirm the inclusion results, and graphical plots illustrate how the operator reshapes the image domains for different parameter pairs

(ν, b)

. Numerical illustrations are provided to visualize the geometric steering effect induced by the Marcum Q-function and its fractional-order damping behavior. 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, 278 KB

Open AccessArticle

Some Geometric Characterizations of a Certain Class of Log-Harmonic Mappings

by Madhusmita Mohanty, Bikash Chinhara and Ram Mohapatra

Mathematics 2026, 14(4), 659; https://doi.org/10.3390/math14040659 - 12 Feb 2026

Viewed by 295

Abstract

This article investigates structural properties of a class of log-harmonic mappings associated with starlike analytic functions in the unit disk. Beginning with a general representation of the log-harmonic mappings, we use sharp inequalities using analytic and dilatation functions to determine growth and distortion [...] Read more.

This article investigates structural properties of a class of log-harmonic mappings associated with starlike analytic functions in the unit disk. Beginning with a general representation of the log-harmonic mappings, we use sharp inequalities using analytic and dilatation functions to determine growth and distortion bounds for the mappings and their complex derivatives. The exact region where the log harmonic mappings of the form

f (z) = z h (z) \bar{h^{'} (z)}

, with h being starlike analytic, give the starlike image is determined. Subordination relations for logarithmic derivatives are established, connecting the mappings with the Schwarz lemma and the Carathéodory class. Furthermore, we obtain the growth estimates for the underlying starlike functions h and their derivatives, as well as accurate inequalities governing the arclength of the circle image under log-harmonic mappings. These findings contribute to the geometric function theory of log-harmonic mappings. Full article

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

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 620

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)

16 pages, 292 KB

Open AccessArticle

Classes of Harmonic Functions Defined by the Carlson–Shaffer Operator

by Jacek Dziok

Symmetry 2025, 17(4), 558; https://doi.org/10.3390/sym17040558 - 6 Apr 2025

Viewed by 857

Abstract

The Carlson–Shaffer operator plays an important role in the geometric theory of analytic functions. It is associated with the hypergeometric function and the incomplete beta function. The Carlson–Shaffer operator generalizes various other linear operators, such as the Ruscheweyh derivative operator, the Bernardi–Libera–Livingston operator, [...] Read more.

The Carlson–Shaffer operator plays an important role in the geometric theory of analytic functions. It is associated with the hypergeometric function and the incomplete beta function. The Carlson–Shaffer operator generalizes various other linear operators, such as the Ruscheweyh derivative operator, the Bernardi–Libera–Livingston operator, and the Srivastava–Owa operator. Ideas in the theory of analytic functions are often symmetrically transferred to the theory of harmonic functions. By using the Carlson–Shaffer operator, we introduce a class of harmonic functions defined by weak subordination. Next, we give some necessary and sufficient coefficient conditions for the class of functions. Furthermore, we determine coefficient estimates, distortion bounds, extreme points, and radii of starlikeness and convexity for the defined class. Full article

(This article belongs to the Section Mathematics)

10 pages, 261 KB

Open AccessArticle

Relations of Harmonic Starlike Function Subclasses with Mittag–Leffler Function

by Naci Taşar, Fethiye Müge Sakar, Seher Melike Aydoğan and Georgia Irina Oros

Axioms 2024, 13(12), 826; https://doi.org/10.3390/axioms13120826 - 26 Nov 2024

Viewed by 838

Abstract

In this study, the connection between certain subfamilies of harmonic univalent functions is established by utilizing a convolution operator involving the Mittag–Leffler function. The investigation reveals inclusion relations concerning harmonic

γ

-uniformly starlike mappings in the open unit disc, harmonic starlike functions and [...] Read more.

In this study, the connection between certain subfamilies of harmonic univalent functions is established by utilizing a convolution operator involving the Mittag–Leffler function. The investigation reveals inclusion relations concerning harmonic

γ

-uniformly starlike mappings in the open unit disc, harmonic starlike functions and harmonic convex functions, highlighting the improvements given by the results presented here on previously published outcomes. Full article

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

17 pages, 293 KB

Open AccessArticle

Starlike Functions in the Space of Meromorphic Harmonic Functions

by Jacek Dziok

Symmetry 2024, 16(9), 1112; https://doi.org/10.3390/sym16091112 - 27 Aug 2024

Cited by 1 | Viewed by 1428

Abstract

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the [...] Read more.

The Geometric Theory of Analytic Functions was initially developed for the space of functions that are analytic in the unit disk. The convexity and starlikeness of functions are the first geometric ideas considered in this theory. We can notice a symmetry between the subjects considered in the space of analytic functions and those in the space of harmonic functions. In the presented paper, we consider the starlikeness of functions in the space of meromorphic harmonic functions. Full article

20 pages, 372 KB

Open AccessArticle

Subclasses of Noshiro-Type Starlike Harmonic Functions Involving q-Srivastava–Attiya Operator

by Gangadharan Murugusundaramoorthy, Kaliappan Vijaya, Daniel Breaz and Luminiţa-Ioana Cotîrlǎ

Mathematics 2023, 11(23), 4711; https://doi.org/10.3390/math11234711 - 21 Nov 2023

Cited by 7 | Viewed by 1311

Abstract

In this paper, the harmonic function related to the q-Srivastava–Attiya operator is described to introduce a new class of complex harmonic functions that are orientation-preserving and univalent in the open-unit disk. We also cover some important aspects such as coefficient bounds, convolution [...] Read more.

In this paper, the harmonic function related to the q-Srivastava–Attiya operator is described to introduce a new class of complex harmonic functions that are orientation-preserving and univalent in the open-unit disk. We also cover some important aspects such as coefficient bounds, convolution conservation, and convexity constraints. Next, using sufficiency criteria, we calculate the sharp bounds of the real parts of the ratios of harmonic functions to their sequences of partial sums. In addition, for the first time some of the interesting implications of the q-Srivastava–Attiya operator in harmonic functions are also included. Full article

14 pages, 394 KB

Open AccessArticle

Certain Properties of Harmonic Functions Defined by a Second-Order Differential Inequality

by Daniel Breaz, Abdullah Durmuş, Sibel Yalçın, Luminita-Ioana Cotirla and Hasan Bayram

Mathematics 2023, 11(19), 4039; https://doi.org/10.3390/math11194039 - 23 Sep 2023

Cited by 4 | Viewed by 1528

Abstract

The Theory of Complex Functions has been studied by many scientists and its application area has become a very wide subject. Harmonic functions play a crucial role in various fields of mathematics, physics, engineering, and other scientific disciplines. Of course, the main reason [...] Read more.

The Theory of Complex Functions has been studied by many scientists and its application area has become a very wide subject. Harmonic functions play a crucial role in various fields of mathematics, physics, engineering, and other scientific disciplines. Of course, the main reason for maintaining this popularity is that it has an interdisciplinary field of application. This makes this subject important not only for those who work in pure mathematics, but also in fields with a deep-rooted history, such as engineering, physics, and software development. In this study, we will examine a subclass of Harmonic functions in the Theory of Geometric Functions. We will give some definitions necessary for this. Then, we will define a new subclass of complex-valued harmonic functions, and their coefficient relations, growth estimates, radius of univalency, radius of starlikeness and radius of convexity of this class are investigated. In addition, it is shown that this class is closed under convolution of its members. Full article

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15 pages, 914 KB

Open AccessArticle

Estimation of the Bounds of Some Classes of Harmonic Functions with Symmetric Conjugate Points

by Lina Ma, Shuhai Li and Huo Tang

Symmetry 2023, 15(9), 1639; https://doi.org/10.3390/sym15091639 - 25 Aug 2023

Cited by 1 | Viewed by 1353

Abstract

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we [...] Read more.

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we discuss the bound of the Bloch constant and the norm of the pre-Schwarzian derivative for the classes. Full article

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

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21 pages, 457 KB

Open AccessArticle

On Some Classes of Harmonic Functions Associated with the Janowski Function

by Lina Ma, Shuhai Li and Huo Tang

Mathematics 2023, 11(17), 3666; https://doi.org/10.3390/math11173666 - 25 Aug 2023

Cited by 2 | Viewed by 2103

Abstract

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, we discuss the geometric properties of the classes, such [...] Read more.

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, we discuss the geometric properties of the classes, such as the integral expression, coefficient estimation, distortion theorem, Jacobian estimation, growth estimates, and covering theorem. Full article

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

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12 pages, 309 KB

Open AccessArticle

Classes of Harmonic Functions Related to Mittag-Leffler Function

by Abeer A. Al-Dohiman, Basem Aref Frasin, Naci Taşar and Fethiye Müge Sakar

Axioms 2023, 12(7), 714; https://doi.org/10.3390/axioms12070714 - 23 Jul 2023

Cited by 4 | Viewed by 1686

Abstract

The purpose of this paper is to find new inclusion relations of the harmonic class

HF (ϱ, γ)

with the subclasses

S_{HF}^{*},

K_{HF}

and

T N_{HF} (τ)

of harmonic functions by applying the [...] Read more.

The purpose of this paper is to find new inclusion relations of the harmonic class

HF (ϱ, γ)

with the subclasses

S_{HF}^{*},

K_{HF}

and

T N_{HF} (τ)

of harmonic functions by applying the convolution operator

Θ (ℑ)

associated with the Mittag-Leffler function. Further for

ϱ = 0

, several special cases of the main results are also obtained. Full article

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

12 pages, 324 KB

Open AccessArticle

Some Properties of Certain Multivalent Harmonic Functions

by Georgia Irina Oros, Sibel Yalçın and Hasan Bayram

Mathematics 2023, 11(11), 2416; https://doi.org/10.3390/math11112416 - 23 May 2023

Cited by 7 | Viewed by 1817

Abstract

In this paper, various features of a new class of normalized multivalent harmonic functions in the open unit disk are analyzed, including bounds on coefficients, growth estimations, starlikeness and convexity radii. It is further demonstrated that this class is closed when its members [...] Read more.

In this paper, various features of a new class of normalized multivalent harmonic functions in the open unit disk are analyzed, including bounds on coefficients, growth estimations, starlikeness and convexity radii. It is further demonstrated that this class is closed when its members are convoluted. It can also be seen that various previously introduced and investigated classes of multivalent harmonic functions appear as special cases for this class. Full article

(This article belongs to the Special Issue New Trends in Complex Analysis Research, 2nd Edition)

11 pages, 295 KB

Open AccessArticle

Studying the Harmonic Functions Associated with Quantum Calculus

by Abdullah Alsoboh, Ala Amourah, Maslina Darus and Carla Amoi Rudder

Mathematics 2023, 11(10), 2220; https://doi.org/10.3390/math11102220 - 9 May 2023

Cited by 19 | Viewed by 2416

Abstract

Using the derivative operators’

q

-analogs values, a wide variety of holomorphic function subclasses,

q

-starlike, and

q

-convex functions have been researched and examined. With the aid of fundamental ideas from the theory of

q

-calculus operators, we describe new

q

-operators [...] Read more.

Using the derivative operators’

q

-analogs values, a wide variety of holomorphic function subclasses,

q

-starlike, and

q

-convex functions have been researched and examined. With the aid of fundamental ideas from the theory of

q

-calculus operators, we describe new

q

-operators of harmonic function

H_{ϱ, χ; q}^{γ} F (ϖ)

in this work. We also define a new harmonic function subclass related to the Janowski and

q

-analog of Le Roy-type functions Mittag–Leffler functions. Several important properties are assigned to the new class, including necessary and sufficient conditions, the covering Theorem, extreme points, distortion bounds, convolution, and convex combinations. Furthermore, we emphasize several established remarks for confirming our primary findings presented in this study, as well as some applications of this study in the form of specific outcomes and corollaries. Full article

(This article belongs to the Special Issue New Trends in Complex Analysis Research, 2nd Edition)

13 pages, 316 KB

Open AccessArticle

New Criteria for Convex-Exponent Product of Log-Harmonic Functions

by Rasoul Aghalary, Ali Ebadian, Nak Eun Cho and Mehri Alizadeh

Axioms 2023, 12(5), 409; https://doi.org/10.3390/axioms12050409 - 22 Apr 2023

Cited by 1 | Viewed by 1497

Abstract

In this study, we consider different types of convex-exponent products of elements of a certain class of log-harmonic mapping and then find sufficient conditions for them to be starlike log-harmonic functions. For instance, we show that, if f is a spirallike function, then [...] Read more.

In this study, we consider different types of convex-exponent products of elements of a certain class of log-harmonic mapping and then find sufficient conditions for them to be starlike log-harmonic functions. For instance, we show that, if f is a spirallike function, then choosing a suitable value of

γ

, the log-harmonic mapping

F (z) = {f (z) | f (z) |}^{2 γ}

is

α

-

s p i r a l i k e

of order

ρ

. Our results generalize earlier work in the literature. Full article

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

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11 pages, 278 KB

Open AccessArticle

An Application of Poisson Distribution Series on Harmonic Classes of Analytic Functions

by Basem Frasin and Alina Alb Lupaş

Symmetry 2023, 15(3), 590; https://doi.org/10.3390/sym15030590 - 24 Feb 2023

Cited by 8 | Viewed by 2279

Abstract

Many authors have obtained some inclusion properties of certain subclasses of univalent and functions associated with distribution series, such as Pascal distribution, Binomial distribution, Poisson distribution, Mittag–Leffler-type Poisson distribution, and Geometric distribution. In the present paper, we obtain some inclusion relations of the [...] Read more.

Many authors have obtained some inclusion properties of certain subclasses of univalent and functions associated with distribution series, such as Pascal distribution, Binomial distribution, Poisson distribution, Mittag–Leffler-type Poisson distribution, and Geometric distribution. In the present paper, we obtain some inclusion relations of the harmonic class

H (α, δ)

with the classes

S_{H}^{*}

of starlike harmonic functions and

K_{H}

of convex harmonic functions, also for the harmonic classes

{TN}_{H} (β)

and

{TR}_{H} (β)

associated with the operator

Υ

defined by applying certain convolution operator regarding Poisson distribution series. Several consequences and corollaries of the main results are also obtained. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

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