MDPI - Publisher of Open Access Journals

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 1392

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)

14 pages, 724 KB

Open AccessArticle

Mapping Properties of Associate Laguerre Polynomial in Symmetric Domains

by Sa’ud Al-Sa’di, Ayesha Siddiqa, Bushra Kanwal, Mohammed Ali Alamri, Saqib Hussain and Saima Noor

Symmetry 2024, 16(11), 1545; https://doi.org/10.3390/sym16111545 - 18 Nov 2024

Cited by 1 | Viewed by 1293

Abstract

The significant characteristics of Associate Laguerre polynomials (ALPs) have noteworthy applications in the fields of complex analysis and mathematical physics. The present article mainly focuses on the inclusion relationships of ALPs and various analytic domains. Starting with the investigation of admissibility conditions of [...] Read more.

The significant characteristics of Associate Laguerre polynomials (ALPs) have noteworthy applications in the fields of complex analysis and mathematical physics. The present article mainly focuses on the inclusion relationships of ALPs and various analytic domains. Starting with the investigation of admissibility conditions of the analytic functions belonging to these domains, we obtained the conditions on the parameters of ALPs under which an ALP maps an open unit disc inside such analytical domains. The graphical demonstration enhances the outcomes and also proves the validity of our obtained results. Full article

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

► Show Figures

Figure 1

19 pages, 297 KB

Open AccessArticle

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

by Hari M. Srivastava, Daniel Breaz, Shahid Khan and Fairouz Tchier

Axioms 2024, 13(6), 366; https://doi.org/10.3390/axioms13060366 - 29 May 2024

Cited by 6 | Viewed by 1824

Abstract

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We [...] Read more.

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

20 pages, 382 KB

Open AccessArticle

New Applications of Fractional q-Calculus Operator for a New Subclass of q-Starlike Functions Related with the Cardioid Domain

by Mohammad Faisal Khan and Mohammed AbaOud

Fractal Fract. 2024, 8(1), 71; https://doi.org/10.3390/fractalfract8010071 - 22 Jan 2024

Cited by 15 | Viewed by 3032

Abstract

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical [...] Read more.

Recently, a number of researchers from different fields have taken a keen interest in the domain of fractional q-calculus on the basis of fractional integrals and derivative operators. This has been used in various scientific research and technology fields, including optics, mathematical biology, plasma physics, electromagnetic theory, and many more. This article explores some mathematical applications of the fractional q-differential and integral operator in the field of geometric function theory. By using the linear multiplier fractional q-differintegral operator

D_{q, λ}^{m} (ρ, σ)

and subordination, we define and develop a collection of q-starlike functions that are linked to the cardioid domain. This study also investigates sharp inequality problems like initial coefficient bounds, the Fekete–Szego problems, and the coefficient inequalities for a new class of q-starlike functions in the open unit disc

U

. Furthermore, we analyze novel findings with respect to the inverse function

(μ^{- 1})

within the class of q-starlike functions in

U

. The findings in this paper are easy to understand and show a connection between present and past studies. Full article

(This article belongs to the Special Issue Advances in Fractional Integral and Derivative Operators with Applications)

15 pages, 372 KB

Open AccessArticle

Third Hankel Determinant for Subclasses of Analytic and m-Fold Symmetric Functions Involving Cardioid Domain and Sine Function

by Ayman Alahmade, Zeeshan Mujahid, Ferdous M. O. Tawfiq, Bilal Khan, Nazar Khan and Fairouz Tchier

Symmetry 2023, 15(11), 2039; https://doi.org/10.3390/sym15112039 - 10 Nov 2023

Cited by 4 | Viewed by 1828

Abstract

In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f [...] Read more.

In this research, we define a few subclasses of analytic functions which are connected to sine functions and the cardioid domain in the unit disk. We investigate initial coefficient bounds, the Fekete–Szego problem and second and third Hankel determinants for the functions f belonging to these newly defined classes. We also define the class of m-fold symmetric functions related with the sine function and then investigate the bounds of the third Hankel determinant for twofold symmetric and threefold symmetric functions. Full article

► Show Figures

Figure 1

22 pages, 357 KB

Open AccessArticle

Investigation of the Hankel Determinant Sharp Bounds for a Specific Analytic Function Linked to a Cardioid-Shaped Domain

by Isra Al-Shbeil, Muhammad Imran Faisal, Muhammad Arif, Muhammad Abbas and Reem K. Alhefthi

Mathematics 2023, 11(17), 3664; https://doi.org/10.3390/math11173664 - 25 Aug 2023

Cited by 3 | Viewed by 1761

Abstract

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these [...] Read more.

One of the challenging tasks in the study of function theory is how to obtain sharp estimates of coefficients that appear in the Taylor–Maclaurin series of analytic univalent functions, and for obtaining these bounds, researchers used the concepts of Carathéodory functions. Among these coefficient-related problems, the problem of the third-order Hankel determinant sharp bound is the most difficult one. The aim of the present study is to determine the sharp bound of the Hankel determinant of third order by using the methodology of the aforementioned Carathéodory function family. Further, we also study some other coefficient-related problems, such as the Fekete–Szegő inequality and the second-order Hankel determinant. We examine these results for the family of bounded turning functions linked with a cardioid-shaped domain. Full article

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

15 pages, 319 KB

Open AccessArticle

Sharp Coefficient Bounds for a Subclass of Bounded Turning Functions with a Cardioid Domain

by Lei Shi, Hari Mohan Srivastava, Nak Eun Cho and Muhammad Arif

Axioms 2023, 12(8), 775; https://doi.org/10.3390/axioms12080775 - 10 Aug 2023

Cited by 9 | Viewed by 1965

Abstract

In the present paper, we give a new simple proof on the sharp bounds of coefficient functionals related to the Carathéodory functions and make a correction on the extremal functions. The result is further used to investigate some initial coefficient bounds on a [...] Read more.

In the present paper, we give a new simple proof on the sharp bounds of coefficient functionals related to the Carathéodory functions and make a correction on the extremal functions. The result is further used to investigate some initial coefficient bounds on a subclass of bounded turning functions

R_{℘}

associated with a cardioid domain. For functions in this class, we calculate the bounds of the Fekete–Szegö-type inequality and the second- and third-order Hankel determinants. All the results are proved to be sharp. Full article

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

22 pages, 351 KB

Open AccessArticle

New Applications of the Sălăgean Quantum Differential Operator for New Subclasses of q-Starlike and q-Convex Functions Associated with the Cardioid Domain

by Suha B. Al-Shaikh

Symmetry 2023, 15(6), 1185; https://doi.org/10.3390/sym15061185 - 1 Jun 2023

Cited by 3 | Viewed by 1757

Abstract

In this paper, we define a new family of q-starlike and q-convex functions related to the cardioid domain utilizing the ideas of subordination and the Sălăgean quantum differential operator. The primary contribution of this article is the derivation of a sharp [...] Read more.

In this paper, we define a new family of q-starlike and q-convex functions related to the cardioid domain utilizing the ideas of subordination and the Sălăgean quantum differential operator. The primary contribution of this article is the derivation of a sharp inequality for the newly established subclasses of q-starlike and q-convex functions in the open unit disc

U

. For this novel family, bounds of the first two Taylor-Maclaurin coefficients, the Fekete-Szegö-type functional, and coefficient inequalities are studied. Furthermore, we also investigate some new results for the inverse function belonging to the classes of q-starlike and q-convex functions. The results presented in this article are sharp. To draw connections between the early and present findings, several well-known corollaries are also highlighted. Symmetric quantum calculus operator theory can be used to investigate the symmetry properties of this new family of functions. Full article

(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)

21 pages, 399 KB

Open AccessArticle

Investigation of the Second-Order Hankel Determinant for Sakaguchi-Type Functions Involving the Symmetric Cardioid-Shaped Domain

by Khalil Ullah, Muhammad Arif, Ibtisam Mohammed Aldawish and Sheza M. El-Deeb

Fractal Fract. 2023, 7(5), 376; https://doi.org/10.3390/fractalfract7050376 - 30 Apr 2023

Viewed by 2017

Abstract

Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of [...] Read more.

Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of problems, such as the first three initial coefficient problems, the Zalcman inequalities, the Fekete–Szegö type results, and the second-order Hankel determinant for families of Sakaguchi-type functions related to the cardioid-shaped domain. Further, we study the logarithmic coefficients for both of these classes. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

20 pages, 334 KB

Open AccessArticle

Sharp Coefficient Bounds for a New Subclass of Starlike Functions of Complex Order γ Associated with Cardioid Domain

by Suha B. Al-Shaikh, Khaled Matarneh, Ahmad A. Abubaker and Mohammad Faisal Khan

Mathematics 2023, 11(9), 2017; https://doi.org/10.3390/math11092017 - 24 Apr 2023

Cited by 2 | Viewed by 1608

Abstract

In this study, by using the concepts of subordination, we define a new family

R (M, N, λ, γ)

of starlike functions of complex order

γ

connected with the cardioid domain. The main contribution of this article consists of the [...] Read more.

In this study, by using the concepts of subordination, we define a new family

R (M, N, λ, γ)

of starlike functions of complex order

γ

connected with the cardioid domain. The main contribution of this article consists of the derivations of sharp inequality, considering the functions belonging to the family

R (M, N, λ, γ)

of starlike functions in

U

. Particularly, sharp bounds of the first two Taylor–Maclaurin coefficients, sharp estimates of the Fekete–Szegö-type functionals, and coefficient inequalities are investigated for this newly defined family

R (M, N, λ, γ)

of starlike functions. Furthermore, for the inverse function and the

log (\frac{g (z)}{z})

function, we investigate the same types of problems. Several well-known corollaries are also highlighted to show the connections between prior research and the new findings. Full article

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

10 pages, 1492 KB

Open AccessArticle

On the Stability Domain of a Class of Linear Systems of Fractional Order

by Marius-F. Danca

Fractal Fract. 2023, 7(1), 49; https://doi.org/10.3390/fractalfract7010049 - 31 Dec 2022

Cited by 7 | Viewed by 2389

Abstract

In this paper, the shape of the stability domain

S^{q}

for a class of difference systems defined by the Caputo forward difference operator

Δ^{q}

of order

q \in (0, 1)

is numerically analyzed. It is shown numerically that [...] Read more.

In this paper, the shape of the stability domain

S^{q}

for a class of difference systems defined by the Caputo forward difference operator

Δ^{q}

of order

q \in (0, 1)

is numerically analyzed. It is shown numerically that due to of power of the negative base in the expression of the stability domain, in addition to the known cardioid-like shapes,

S^{q}

could present supplementary regions where the stability is not verified. The Mandelbrot map of fractional order is considered as an illustrative example. In addition, it is conjectured that for

q < 0.5

, the shape of

S^{q}

does not cover the main body of the underlying Mandelbrot set of fractional order as in the case of integer order. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Embedded Systems)

► Show Figures

Figure 1

12 pages, 346 KB

Open AccessArticle

On Convex Functions Associated with Symmetric Cardioid Domain

by Sarfraz Nawaz Malik, Mohsan Raza, Qin Xin, Janusz Sokół, Rabbiya Manzoor and Saira Zainab

Symmetry 2021, 13(12), 2321; https://doi.org/10.3390/sym13122321 - 4 Dec 2021

Cited by 13 | Viewed by 2890

Abstract

The geometry of the image domain plays an important role in the characterization of analytic functions. Therefore, for a comprehensive and detailed study of these functions, a thorough analysis of the geometrical properties of their domains is of prime interest. In this regard, [...] Read more.

The geometry of the image domain plays an important role in the characterization of analytic functions. Therefore, for a comprehensive and detailed study of these functions, a thorough analysis of the geometrical properties of their domains is of prime interest. In this regard, new geometrical structures are introduced and studied as an image domain and then their subsequent analytic functions are defined. Inspired and motivated by ongoing research, Malik et al. introduced a very innovative domain named the cardioid domain, which is symmetric about a real axis. Extending the same work on this symmetric cardioid domain, in this article, we provide a deeper analysis and define and study the convex functions associated with the symmetric cardioid domain, named cardio-convex functions. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

► Show Figures

Figure 1

20 pages, 1204 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Discrete-Time Pole-Region Robust Controller for Magnetic Levitation Plant

by Mária Hypiusová and Danica Rosinová

Symmetry 2021, 13(1), 142; https://doi.org/10.3390/sym13010142 - 16 Jan 2021

Cited by 11 | Viewed by 3742

Abstract

Robust pole-placement based on convex

D_{R}

-regions belongs to the efficient control design techniques for real systems, providing computationally tractable pole-placement design algorithms. The problem arises in the discrete-time domain when the relative damping is prescribed since the corresponding discrete-time domain is [...] Read more.

Robust pole-placement based on convex

D_{R}

-regions belongs to the efficient control design techniques for real systems, providing computationally tractable pole-placement design algorithms. The problem arises in the discrete-time domain when the relative damping is prescribed since the corresponding discrete-time domain is non-convex, having a “cardioid” shape. In this paper, we further develop our recent results on the inner convex approximations of the cardioid, present systematical analysis of its design parameters and their influence on the corresponding closed loop performance (measured by standard integral of absolute error (IAE) and Total Variance criteria). The application of a robust controller designed with the proposed convex approximation of the discrete-time pole region is illustrated and evaluated on a real laboratory magnetic levitation plant. Full article

(This article belongs to the Special Issue PID Control and Symmetry)

► Show Figures

Figure 1

16 pages, 308 KB

Open AccessArticle

q-Analogue of Differential Subordinations

by Miraj Ul-Haq, Mohsan Raza, Muhammad Arif, Qaiser Khan and Huo Tang

Mathematics 2019, 7(8), 724; https://doi.org/10.3390/math7080724 - 9 Aug 2019

Cited by 39 | Viewed by 3705

Abstract

In this article, we study differential subordnations in q-analogue. Some properties of analytic functions in q-analogue associated with cardioid domain and limacon domain are considered. In particular, we determine conditions on

α

such that [...] Read more.

In this article, we study differential subordnations in q-analogue. Some properties of analytic functions in q-analogue associated with cardioid domain and limacon domain are considered. In particular, we determine conditions on

α

such that

1 + α \frac{z \partial_{q} h (z)}{{(h (z))}^{n}} (n = 0, 1, 2, 3)

are subordinated by Janowski functions and

h (z) ≺ 1 + \frac{4}{3} z + \frac{2}{3} z^{2} .

We also consider the same implications such that

h (z) ≺ 1 + \sqrt{2} z + \frac{1}{2} z^{2}

. We apply these results on analytic functions to find sufficient conditions for q-starlikeness related with cardioid and limacon. Full article

(This article belongs to the Special Issue Fractional-Order Integral and Derivative Operators and Their Applications)

15 pages, 771 KB

Open AccessArticle

A Study of Third Hankel Determinant Problem for Certain Subfamilies of Analytic Functions Involving Cardioid Domain

by Lei Shi, Izaz Ali, Muhammad Arif, Nak Eun Cho, Shehzad Hussain and Hassan Khan

Mathematics 2019, 7(5), 418; https://doi.org/10.3390/math7050418 - 10 May 2019

Cited by 46 | Viewed by 3904

Abstract

In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the [...] Read more.

In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the same bounds have been investigated for two-fold and three-fold symmetric functions. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)

Search Results (15)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (15)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI