Research

16 pages, 309 KiB

Open AccessArticle

by Xuewu Zuo, Saad Ihsan Butt, Muhammad Umar, Hüseyin Budak and Muhammad Aamir Ali

Symmetry 2023, 15(8), 1576; https://doi.org/10.3390/sym15081576 - 12 Aug 2023

Cited by 1 | Viewed by 652

The striking goal of this study is to introduce a

q

-identity for a parameterized integral operator via differentiable function. First, we discover the parameterized lemma for the

q

-integral. After that, we provide several

q

-differentiable inequalities. By taking suitable choices, some [...] Read more.

The striking goal of this study is to introduce a

q

-identity for a parameterized integral operator via differentiable function. First, we discover the parameterized lemma for the

q

-integral. After that, we provide several

q

-differentiable inequalities. By taking suitable choices, some interesting results are obtained. With all of these, we displayed the findings from the traditional analysis utilizing

q \to 1^{-}

. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

20 pages, 506 KiB

Open AccessArticle

New Perspectives of Symmetry Conferred by q-Hermite-Hadamard Type Integral Inequalities

by Loredana Ciurdariu and Eugenia Grecu

Symmetry 2023, 15(8), 1514; https://doi.org/10.3390/sym15081514 - 31 Jul 2023

Cited by 1 | Viewed by 701

Abstract

The main goal of this work is to provide quantum parametrized Hermite-Hadamard like type integral inequalities for functions whose second quantum derivatives in absolute values follow different type of convexities. A new quantum integral identity is derived for twice quantum differentiable functions, which [...] Read more.

The main goal of this work is to provide quantum parametrized Hermite-Hadamard like type integral inequalities for functions whose second quantum derivatives in absolute values follow different type of convexities. A new quantum integral identity is derived for twice quantum differentiable functions, which is used as a key element in our demonstrations along with several basic inequalities such as: power mean inequality, and Holder’s inequality. The symmetry of the Hermite-Hadamard type inequalities is stressed by the different types of convexities. Several special cases of the parameter are chosen to illustrate the investigated results. Four examples are presented. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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31 pages, 421 KiB

Open AccessArticle

Raina’s Function-Based Formulations of Right-Sided Simpson’s and Newton’s Inequalities for Generalized Coordinated Convex Functions

by Miguel Vivas-Cortez, Ghulam Murtaza, Ghulam Murtaza Baig and Muhammad Uzair Awan

Symmetry 2023, 15(7), 1441; https://doi.org/10.3390/sym15071441 - 18 Jul 2023

Cited by 1 | Viewed by 1122

Abstract

The main focus of this article is to derive some new counterparts to Simpson’s and Newton’s type inequalities involve a class of generalized coordinated convex mappings. This class contains several new and known classes of convexity as special cases. For further demonstration, we [...] Read more.

The main focus of this article is to derive some new counterparts to Simpson’s and Newton’s type inequalities involve a class of generalized coordinated convex mappings. This class contains several new and known classes of convexity as special cases. For further demonstration, we deploy the concept of right quantum derivatives to develop two new identities involving Raina’s function. Moreover, by implementing these auxiliary results together with generalized convexity, we acquire a Holder-type inequality. We also acquire some applications of our main findings by making use of suitable substitutions in Raina’s function. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

13 pages, 2731 KiB

Open AccessArticle

Numerical Analysis of Nonlinear Fractional System of Jaulent–Miodek Equation

by Abdulrahman A. B. M. Alzahrani

Symmetry 2023, 15(7), 1350; https://doi.org/10.3390/sym15071350 - 03 Jul 2023

Cited by 1 | Viewed by 772

Abstract

This paper presents the optimal auxiliary function method (OAFM) implementation to solve a nonlinear fractional system of the Jaulent–Miodek Equation with the Caputo operator. The OAFM is a vital method for solving different kinds of nonlinear equations. In this paper, the OAFM is [...] Read more.

This paper presents the optimal auxiliary function method (OAFM) implementation to solve a nonlinear fractional system of the Jaulent–Miodek Equation with the Caputo operator. The OAFM is a vital method for solving different kinds of nonlinear equations. In this paper, the OAFM is applied to the fractional nonlinear system of the Jaulent–Miodek Equation, which describes the behavior of a physical system via a set of coupled nonlinear equations. The Caputo operator represents the fractional derivative in the equations, improving the system’s accuracy and applicability to the real world. This study demonstrates the effectiveness and efficiency of the OAFM in solving the fractional nonlinear system of the Jaulent–Miedek equation with the Caputo operator. This study’s findings provide important insights into the behavior of complex physical systems and may have practical applications in fields such as engineering, physics, and mathematics. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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12 pages, 323 KiB

Open AccessArticle

Exploring Quantum Simpson-Type Inequalities for Convex Functions: A Novel Investigation

by Sabah Iftikhar, Muhammad Uzair Awan and Hüseyin Budak

Symmetry 2023, 15(7), 1312; https://doi.org/10.3390/sym15071312 - 27 Jun 2023

Viewed by 735

Abstract

This study seeks to derive novel quantum variations of Simpson’s inequality by primarily utilizing the convexity characteristics of functions. Additionally, the study examines the credibility of the obtained results through the presentation of relevant numerical examples and graphs. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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20 pages, 4383 KiB

Open AccessArticle

Some q-Symmetric Integral Inequalities Involving s-Convex Functions

by Ammara Nosheen, Sana Ijaz, Khuram Ali Khan, Khalid Mahmood Awan, Marwan Ali Albahar and Mohammed Thanoon

Symmetry 2023, 15(6), 1169; https://doi.org/10.3390/sym15061169 - 29 May 2023

Cited by 4 | Viewed by 890

Abstract

The q-symmetric analogues of Hölder, Minkowski, and power mean inequalities are presented in this paper. The obtained inequalities along with a Montgomery identity involving q-symmetric integrals are used to extend some Ostrowski-type inequalities. The q-symmetric derivatives of the functions involved [...] Read more.

The q-symmetric analogues of Hölder, Minkowski, and power mean inequalities are presented in this paper. The obtained inequalities along with a Montgomery identity involving q-symmetric integrals are used to extend some Ostrowski-type inequalities. The q-symmetric derivatives of the functions involved in these Ostrowski-type inequalities are convex or s-convex. Moreover, some Hermite–Hadamard inequalities for convex functions as well as for s-convex functions are also acquired with the help of q-symmetric calculus in the present work. Some examples are included to support the effectiveness of the proved results. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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16 pages, 359 KiB

Open AccessArticle

Exploration of Quantum Milne–Mercer-Type Inequalities with Applications

by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Awais Gul Khan, Clemente Cesarano and Muhammad Aslam Noor

Symmetry 2023, 15(5), 1096; https://doi.org/10.3390/sym15051096 - 16 May 2023

Cited by 4 | Viewed by 1072

Abstract

Quantum calculus provides a significant generalization of classical concepts and overcomes the limitations of classical calculus in tackling non-differentiable functions. Implementing the q-concepts to obtain fresh variants of classical outcomes is a very intriguing aspect of research in mathematical analysis. The objective [...] Read more.

Quantum calculus provides a significant generalization of classical concepts and overcomes the limitations of classical calculus in tackling non-differentiable functions. Implementing the q-concepts to obtain fresh variants of classical outcomes is a very intriguing aspect of research in mathematical analysis. The objective of this article is to establish novel Milne-type integral inequalities through the utilization of the Mercer inequality for q-differentiable convex mappings. In order to accomplish this task, we begin by demonstrating a new quantum identity of the Milne type linked to left and right q derivatives. This serves as a supporting result for our primary findings. Our approach involves using the q-equality, well-known inequalities, and convex mappings to obtain new error bounds of the Milne–Mercer type. We also provide some special cases, numerical examples, and graphical analysis to evaluate the efficacy of our results. To the best of our knowledge, this is the first article to focus on quantum Milne–Mercer-type inequalities and we hope that our methods and findings inspire readers to conduct further investigation into this problem. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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19 pages, 382 KiB

Open AccessArticle

New Study on the Quantum Midpoint-Type Inequalities for Twice q-Differentiable Functions via the Jensen–Mercer Inequality

by Saad Ihsan Butt, Muhammad Umar and Hüseyin Budak

Symmetry 2023, 15(5), 1038; https://doi.org/10.3390/sym15051038 - 08 May 2023

Cited by 1 | Viewed by 1092

Abstract

The objective of this study is to identify novel quantum midpoint-type inequalities for twice

q

-differentiable functions by utilizing Mercer’s approach. We introduce a new auxiliary variant of the quantum Mercer midpoint-type identity related to twice

q

-differentiable functions. By applying the theory [...] Read more.

The objective of this study is to identify novel quantum midpoint-type inequalities for twice

q

-differentiable functions by utilizing Mercer’s approach. We introduce a new auxiliary variant of the quantum Mercer midpoint-type identity related to twice

q

-differentiable functions. By applying the theory of convex functions to this identity, we introduce new bounds using well-known inequalities, such as H"older’s inequality and power-mean inequality. We provide explicit examples along with graphical demonstrations. The findings of this study explain previous studies on midpoint-type inequalities. Analytic inequalities of this type, as well as related strategies, have applications in various fields where symmetry plays an important role. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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13 pages, 1136 KiB

Open AccessArticle

Investigating the Impact of Fractional Non-Linearity in the Klein–Fock–Gordon Equation on Quantum Dynamics

by Saima Noor, Azzh Saad Alshehry, Noufe H. Aljahdaly, Hina M. Dutt, Imran Khan and Rasool Shah

Symmetry 2023, 15(4), 881; https://doi.org/10.3390/sym15040881 - 07 Apr 2023

Cited by 3 | Viewed by 1173

Abstract

In this paper, we investigate the fractional-order Klein–Fock–Gordon equations on quantum dynamics using a new iterative method and residual power series method based on the Caputo operator. The fractional-order Klein–Fock–Gordon equation is a generalization of the traditional Klein–Fock–Gordon equation that allows for non-integer [...] Read more.

In this paper, we investigate the fractional-order Klein–Fock–Gordon equations on quantum dynamics using a new iterative method and residual power series method based on the Caputo operator. The fractional-order Klein–Fock–Gordon equation is a generalization of the traditional Klein–Fock–Gordon equation that allows for non-integer orders of differentiation. This equation has been used in the study of quantum dynamics to model the behavior of particles with fractional spin. The Laplace transform is employed to transform the equations into a simpler form, and the resulting equations are then solved using the proposed methods. The accuracy and efficiency of the method are demonstrated through numerical simulations, which show that the method is superior to existing numerical methods in terms of accuracy and computational time. The proposed method is applicable to a wide range of fractional-order differential equations, and it is expected to find applications in various areas of science and engineering. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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13 pages, 460 KiB

Open AccessArticle

Investigating the Dynamics of Time-Fractional Drinfeld–Sokolov–Wilson System through Analytical Solutions

by Saima Noor, Azzh Saad Alshehry, Hina M. Dutt, Robina Nazir, Asfandyar Khan and Rasool Shah

Symmetry 2023, 15(3), 703; https://doi.org/10.3390/sym15030703 - 11 Mar 2023

Cited by 3 | Viewed by 1110

Abstract

This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined to tackle the problem. The Caputo manner is [...] Read more.

This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined to tackle the problem. The Caputo manner is used to describe fractional derivative, and He’s polynomials and Adomian polynomials are employed to address nonlinearity. By following these approaches, we obtain solutions in the form of convergent series. We verify and demonstrate the effectiveness of our suggested strategies by examining the assumed model in terms of fractional order. We use plots for various fractional orders to represent the physical behavior of the suggested technique solutions, and show a numerical simulation. The results demonstrate that the suggested algorithms are systematic, simple to use, effective, and accurate in analyzing the behavior of coupled nonlinear differential equations of fractional order in related scientific and engineering fields. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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14 pages, 613 KiB

Open AccessArticle

Numerical Analysis of Fractional-Order Camassa–Holm and Degasperis–Procesi Models

by Meshari Alesemi

Symmetry 2023, 15(2), 269; https://doi.org/10.3390/sym15020269 - 18 Jan 2023

Cited by 1 | Viewed by 1048

Abstract

This study proposes innovative methods for the time-fractional modified Degasperis–Procesi (mDP) and Camassa–Holm (mCH) models of solitary wave solutions. To formulate the concepts of the homotopy perturbation transform method (HPTM) and Elzaki transform decomposition method (ETDM), we mix the Elzaki transform (ET), homotopy [...] Read more.

This study proposes innovative methods for the time-fractional modified Degasperis–Procesi (mDP) and Camassa–Holm (mCH) models of solitary wave solutions. To formulate the concepts of the homotopy perturbation transform method (HPTM) and Elzaki transform decomposition method (ETDM), we mix the Elzaki transform (ET), homotopy perturbation method (HPM), and Adomian decomposition method (ADM). The Caputo sense is applied to this work. The solutions to a few numerical examples of the modified Degasperis–Procesi (mDP) and Camassa–Holm (mCH) are shown for integer and fractional orders of the issues. The derived and precise solutions are compared using two-dimensional and three-dimensional plots of the solutions, confirming the suggested method’s improved accuracy. Tables are created for each problem to display the suggested approach’s results, precise solutions, and absolute error. These methods provide the iterations as a series of solutions. To show the proposed techniques’ efficiency, we compute the absolute error. It is evident from the estimated values that the approaches are precise and simple and that they can therefore be further extended to linear and nonlinear issues. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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13 pages, 310 KiB

Open AccessArticle

Post-Quantum Integral Inequalities for Three-Times (p,q)-Differentiable Functions

by Loredana Ciurdariu and Eugenia Grecu

Symmetry 2023, 15(1), 246; https://doi.org/10.3390/sym15010246 - 16 Jan 2023

Cited by 2 | Viewed by 1160

Abstract

A new

(p, q)

-integral identity involving left and right post quantum derivatives, by using three times

(p, q)

-differentiable functions is established and then this identity is used to derive several new post-quantum Ostrowski type integral [...] Read more.

A new

(p, q)

-integral identity involving left and right post quantum derivatives, by using three times

(p, q)

-differentiable functions is established and then this identity is used to derive several new post-quantum Ostrowski type integral inequalities for three times

(p, q)

-differentiable functions. These results are generalizations of corresponding results in the area of integral inequalities. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

19 pages, 1037 KiB

Open AccessArticle

Numerical Analysis of Fractional-Order Parabolic Equation Involving Atangana–Baleanu Derivative

by Meshari Alesemi

Symmetry 2023, 15(1), 237; https://doi.org/10.3390/sym15010237 - 15 Jan 2023

Cited by 1 | Viewed by 1020

Abstract

In this study, the suggested q-homotopy analysis transform method is used to compute a numerical solution of a fractional parabolic equation, and the solution is obtained in a fast convergent series. The leverage and efficacy of the suggested technique are demonstrated by the [...] Read more.

In this study, the suggested q-homotopy analysis transform method is used to compute a numerical solution of a fractional parabolic equation, and the solution is obtained in a fast convergent series. The leverage and efficacy of the suggested technique are demonstrated by the test examples provided. The results that were acquired are graphically displayed. The series solution in a sizable admissible domain is handled in an extreme way by the current method. It provides us with a simple means of modifying the solution’s convergence zone. The effectiveness and potential of the suggested algorithm are explicitly shown in the results using graphs. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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14 pages, 663 KiB

Open AccessArticle

A Comparative Study of Fractional Partial Differential Equations with the Help of Yang Transform

by Muhammad Naeem, Humaira Yasmin, Rasool Shah, Nehad Ali Shah and Jae Dong Chung

Symmetry 2023, 15(1), 146; https://doi.org/10.3390/sym15010146 - 04 Jan 2023

Cited by 14 | Viewed by 2195

Abstract

In applied sciences and engineering, partial differential equations (PDE) of integer and non-integer order play a crucial role. It can be challenging to determine these equations’ exact solutions. As a result, developing numerical approaches to obtain precise numerical solutions to these kinds of [...] Read more.

In applied sciences and engineering, partial differential equations (PDE) of integer and non-integer order play a crucial role. It can be challenging to determine these equations’ exact solutions. As a result, developing numerical approaches to obtain precise numerical solutions to these kinds of differential equations takes time. The homotopy perturbation transform method (HPTM) and Yang transform decomposition method (YTDM) are the subjects of several recent findings that we describe. These techniques work well for fractional calculus applications. We also examine fractional differential equations’ precise and approximative solutions. The Caputo derivative is employed because it enables the inclusion of traditional initial and boundary conditions in the formulation of the issue. This has major implications for complicated problems. The paper lists the important characteristics of the YTDM and HPTM. Our research has numerous applications in the disciplines of science and engineering and might be seen as a substitute for current methods. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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13 pages, 306 KiB

Open AccessArticle

On a Certain Subclass of p-Valent Analytic Functions Involving q-Difference Operator

by Abdel Moneim Y. Lashin, Abeer O. Badghaish and Badriah Maeed Algethami

Symmetry 2023, 15(1), 93; https://doi.org/10.3390/sym15010093 - 29 Dec 2022

Cited by 3 | Viewed by 1028

Abstract

This paper introduces and studies a new class of analytic p-valent functions in the open symmetric unit disc involving the Sălăgean-type q-difference operator. Furthermore, we present several interesting subordination results, coefficient inequalities, fractional q-calculus applications, and distortion theorems. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

16 pages, 337 KiB

Open AccessArticle

Geometric Properties for a New Class of Analytic Functions Defined by a Certain Operator

by Daniel Breaz, Gangadharan Murugusundaramoorthy and Luminiţa-Ioana Cotîrlǎ

Symmetry 2022, 14(12), 2624; https://doi.org/10.3390/sym14122624 - 11 Dec 2022

Cited by 4 | Viewed by 1017

Abstract

The aim of this paper is to define and explore a certain class of analytic functions involving the

(p, q)

-Wanas operator related to the Janowski functions. We discuss geometric properties, growth and distortion bounds, necessary and sufficient conditions, the [...] Read more.

The aim of this paper is to define and explore a certain class of analytic functions involving the

(p, q)

-Wanas operator related to the Janowski functions. We discuss geometric properties, growth and distortion bounds, necessary and sufficient conditions, the Fekete–Szegö problem, partial sums, and convex combinations for the newly defined class. We solve the Fekete–Szegö problem related to the convolution product and discuss applications to probability distribution. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

27 pages, 697 KiB

Open AccessArticle

A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus

by V. Rexma Sherine, T. G. Gerly, P. Chellamani, Esmail Hassan Abdullatif Al-Sabri, Rashad Ismail, G. Britto Antony Xavier and N. Avinash

Symmetry 2022, 14(12), 2604; https://doi.org/10.3390/sym14122604 - 08 Dec 2022

Viewed by 1039

Abstract

In this paper, we develop theorems on finite and infinite summation formulas by utilizing the

q

and

(q, h)

anti-difference operators, and also we extend these core theorems to

q_{(α)}

and

{(q, h)}_{α}

[...] Read more.

In this paper, we develop theorems on finite and infinite summation formulas by utilizing the

q

and

(q, h)

anti-difference operators, and also we extend these core theorems to

q_{(α)}

and

{(q, h)}_{α}

difference operators. Several integer order theorems based on

q

and

q_{(α)}

difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for

q

and

q_{(α)}

difference operators. In order to develop the fractional order anti-difference equations for

q

and

q_{(α)}

difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for

q

and

q_{(α)}

difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the

(q, h)

and

{(q, h)}_{α}

difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the

(q, h)

and

{(q, h)}_{α}

difference operators for verification. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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11 pages, 279 KiB

Open AccessArticle

Positive Solutions for a Class of Integral Boundary Value Problem of Fractional q-Difference Equations

by Shugui Kang, Yunfang Zhang, Huiqin Chen and Wenying Feng

Symmetry 2022, 14(11), 2465; https://doi.org/10.3390/sym14112465 - 21 Nov 2022

Cited by 2 | Viewed by 1230

Abstract

This paper studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green’s function and obtain an important property of the function. The new property allows us to prove sufficient conditions [...] Read more.

This paper studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green’s function and obtain an important property of the function. The new property allows us to prove sufficient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a fixed point theorem on order intervals. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

22 pages, 360 KiB

Open AccessArticle

On q-Limaçon Functions

by Afis Saliu, Kanwal Jabeen, Isra Al-Shbeil, Najla Aloraini and Sarfraz Nawaz Malik

Symmetry 2022, 14(11), 2422; https://doi.org/10.3390/sym14112422 - 15 Nov 2022

Cited by 8 | Viewed by 1100

Abstract

Very recently, functions that map the open unit disc U onto a limaçon domain, which is symmetric with respect to the real axis in the right-half plane, were initiated in the literature. The analytic characterization, geometric properties, and Hankel determinants of these families [...] Read more.

Very recently, functions that map the open unit disc U onto a limaçon domain, which is symmetric with respect to the real axis in the right-half plane, were initiated in the literature. The analytic characterization, geometric properties, and Hankel determinants of these families of functions were also demonstrated. In this article, we present a q-analogue of these functions and use it to establish the classes of starlike and convex limaçon functions that are correlated with q-calculus. Furthermore, the coefficient bounds, as well as the third Hankel determinants, for these novel classes are established. Moreover, at some stages, the radius of the inclusion relationship for a particular case of these subclasses with the Janowski families of functions are obtained. It is worth noting that many of our results are sharp. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

19 pages, 521 KiB

Open AccessArticle

Common Fixed Point Theorems on Orthogonal Branciari Metric Spaces with an Application

by Gunaseelan Mani, Senthil Kumar Prakasam, Arul Joseph Gnanaprakasam, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby, Khizar Hyatt Khan and Stojan Radenović

Symmetry 2022, 14(11), 2420; https://doi.org/10.3390/sym14112420 - 15 Nov 2022

Cited by 4 | Viewed by 993

Abstract

In this article, we modify the symmetry of orthogonal metric spaces and we prove common fixed point theorems via simulation functions in orthogonal Rectangular metric spaces. We also provide an illustrative example to support our results. The derived results have been applied to [...] Read more.

In this article, we modify the symmetry of orthogonal metric spaces and we prove common fixed point theorems via simulation functions in orthogonal Rectangular metric spaces. We also provide an illustrative example to support our results. The derived results have been applied to find analytical solutions to integral equations. The analytical solutions are verified with a numerical simulation. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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16 pages, 316 KiB

Open AccessArticle

Applications of Symmetric Quantum Calculus to the Class of Harmonic Functions

by Mohammad Faisal Khan, Isra Al-Shbeil, Najla Aloraini, Nazar Khan and Shahid Khan

Symmetry 2022, 14(10), 2188; https://doi.org/10.3390/sym14102188 - 18 Oct 2022

Cited by 7 | Viewed by 1106

Abstract

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work [...] Read more.

In the past few years, many scholars gave much attention to the use of q-calculus in geometric functions theory, and they defined new subclasses of analytic and harmonic functions. While using the symmetric q-calculus in geometric function theory, very little work has been published so far. In this research, with the help of fundamental concepts of symmetric q-calculus and the symmetric q-Salagean differential operator for harmonic functions, we define a new class of harmonic functions connected with Janowski functions

\tilde{S_{H}^{0}} (m, q, A, B)

. First, we illustrate the necessary and sufficient convolution condition for

\tilde{S_{H}^{0}} (m, q, A, B)

and then prove that this sufficient condition is a sense preserving and univalent, and it is necessary for its subclass

\tilde{{TS}_{H}^{0}} (m, q, A, B)

. Furthermore, by using this necessary and sufficient coefficient condition, we establish some novel results, particularly convexity, compactness, radii of q-starlike and q-convex functions of order

α

, and extreme points for this newly defined class of harmonic functions. Our results are the generalizations of some previous known results. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

12 pages, 388 KiB

Open AccessArticle

Quantumness’ Degree of Thermal Optics’ Approximations

by Flavia Pennini and Angelo Plastino

Symmetry 2022, 14(10), 2052; https://doi.org/10.3390/sym14102052 - 01 Oct 2022

Cited by 1 | Viewed by 802

Abstract

We assess the degree of quantumness of the P, Q, and W quantum optics’ approximations in a thermal context governed by the canonical ensemble treatment. First, we remint the reader of the bridge connecting quantum optics with statistical mechanics using the [...] Read more.

We assess the degree of quantumness of the P, Q, and W quantum optics’ approximations in a thermal context governed by the canonical ensemble treatment. First, we remint the reader of the bridge connecting quantum optics with statistical mechanics using the abovementioned approximations at the temperature T. With the ensuing materials, we explore with some detail some features of the above bridge, related to the entropy and to thermal uncertainties. Some new relations concerning the degree of quantumness of the P, Q, and W are obtained by comparison between them and the exact and classical treatments. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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23 pages, 347 KiB

Open AccessArticle

Estimation of the Second-Order Hankel Determinant of Logarithmic Coefficients for Two Subclasses of Starlike Functions

by Pongsakorn Sunthrayuth, Ibtisam Aldawish, Muhammad Arif, Muhammad Abbas and Sheza El-Deeb

Symmetry 2022, 14(10), 2039; https://doi.org/10.3390/sym14102039 - 29 Sep 2022

Cited by 6 | Viewed by 1157

Abstract

In our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first four initial logarithmic coefficients, the [...] Read more.

In our present study, two subclasses of starlike functions which are symmetric about the origin are considered. These two classes are defined with the use of the sigmoid function and the trigonometric function, respectively. We estimate the first four initial logarithmic coefficients, the Zalcman functional, the Fekete–Szegö functional, and the bounds of second-order Hankel determinants with logarithmic coefficients for the first class

S_{seg}^{*}

and improve the obtained estimate of the existing second-order Hankel determinant of logarithmic coefficients for the second class

S_{sin}^{*}

. All the bounds that we obtain in this article are proven to be sharp. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

21 pages, 425 KiB

Open AccessArticle

New Quantum Mercer Estimates of Simpson–Newton-like Inequalities via Convexity

by Saad Ihsan Butt, Hüseyin Budak and Kamsing Nonlaopon

Symmetry 2022, 14(9), 1935; https://doi.org/10.3390/sym14091935 - 16 Sep 2022

Cited by 8 | Viewed by 1201

Abstract

Recently, developments and extensions of quadrature inequalities in quantum calculus have been extensively studied. As a result, several quantum extensions of Simpson’s and Newton’s estimates are examined in order to explore different directions in quantum studies. The main motivation of this article is [...] Read more.

Recently, developments and extensions of quadrature inequalities in quantum calculus have been extensively studied. As a result, several quantum extensions of Simpson’s and Newton’s estimates are examined in order to explore different directions in quantum studies. The main motivation of this article is the development of variants of Simpson–Newton-like inequalities by employing Mercer’s convexity in the context of quantum calculus. The results also give new quantum bounds for Simpson–Newton-like inequalities through Hölder’s inequality and the power mean inequality by employing the Mercer scheme. The validity of our main results is justified by providing examples with graphical representations thereof. The obtained results recapture the discoveries of numerous authors in quantum and classical calculus. Hence, the results of these inequalities lead us to the development of new perspectives and extensions of prior results. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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19 pages, 508 KiB

Open AccessArticle

Quantum Integral Inequalities in the Setting of Majorization Theory and Applications

by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Hüseyin Budak, Hasan Kara and Muhammad Aslam Noor

Symmetry 2022, 14(9), 1925; https://doi.org/10.3390/sym14091925 - 14 Sep 2022

Cited by 3 | Viewed by 1194

Abstract

In recent years, the theory of convex mappings has gained much more attention due to its massive utility in different fields of mathematics. It has been characterized by different approaches. In 1929, G. H. Hardy, J. E. Littlewood, and G. Polya established another [...] Read more.

In recent years, the theory of convex mappings has gained much more attention due to its massive utility in different fields of mathematics. It has been characterized by different approaches. In 1929, G. H. Hardy, J. E. Littlewood, and G. Polya established another characterization of convex mappings involving an ordering relationship defined over

R^{n}

known as majorization theory. Using this theory many inequalities have been obtained in the literature. In this paper, we study Hermite–Hadamard type inequalities using the Jensen–Mercer inequality in the frame of

\underset{˙}{q}

-calculus and majorized l-tuples. Firstly we derive

\underset{˙}{q}

-Hermite–Hadamard–Jensen–Mercer (H.H.J.M) type inequalities with the help of Mercer’s inequality and its weighted form. To obtain some new generalized (H.H.J.M)-type inequalities, we prove a generalized quantum identity for

\underset{˙}{q}

-differentiable mappings. Next, we obtain some estimation-type results; for this purpose, we consider

\underset{˙}{q}

-identity, fundamental inequalities and the convexity property of mappings. Later on, We offer some applications to special means that demonstrate the importance of our main results. With the help of numerical examples, we also check the validity of our main outcomes. Along with this, we present some graphical analyses of our main results so that readers may easily grasp the results of this paper. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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15 pages, 322 KiB

Open AccessArticle

The Range of a Module Measure Defined on an Effect Algebra

by Francisco Javier García-Pacheco

Symmetry 2022, 14(9), 1819; https://doi.org/10.3390/sym14091819 - 02 Sep 2022

Cited by 2 | Viewed by 869

Abstract

Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an [...] Read more.

Effect algebras are the main object of study in quantum mechanics. Module measures are those measures defined on an effect algebra with values on a topological module. Let R be a topological ring and M a topological R-module. Let L be an effect algebra. The range of a module measure

μ : L \to M

is studied. Among other results, we prove that if L is an sRDP

σ

-effect algebra with a natural basis and

μ : L \to R

is a countably additive measure, then

μ

has bounded variation. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

11 pages, 306 KiB

Open AccessArticle

Discrete Dynamic Model of a Disease-Causing Organism Caused by 2D-Quantum Tsallis Entropy

by Nadia M. G. Al-Saidi, Husam Yahya and Suzan J. Obaiys

Symmetry 2022, 14(8), 1677; https://doi.org/10.3390/sym14081677 - 12 Aug 2022

Cited by 1 | Viewed by 1098

Abstract

Many aspects of the asymmetric organ system are controlled by the symmetry model (R&L) of the disease-causing organism pathway, but sensitive matters like somites and limb buds need to be shielded from its influence. Because symmetric and asymmetric structures develop from similar or [...] Read more.

Many aspects of the asymmetric organ system are controlled by the symmetry model (R&L) of the disease-causing organism pathway, but sensitive matters like somites and limb buds need to be shielded from its influence. Because symmetric and asymmetric structures develop from similar or nearby matters and utilize many of the same signaling pathways, attaining symmetry is made more difficult. On this note, we aim to generalize some important measurements in view of the 2D-quantum calculus (q-calculus, q-analogues or q-disease), including the dimensional of fractals and Tsallis entropy (2D-quantum Tsallis entropy (2D-QTE)). The process is based on producing a generalization of the maximum value of the Tsallis entropy in view of the quantum calculus. Then by considering the maximum 2D-QTE, we design a discrete system. As an application, by using the 2D-QTE, we depict a discrete dynamic system that is afflicted with a disease-causing organism (DCO). We look at the system’s positive and maximum solutions. Studies are done on equilibrium and stability. We will also develop a novel design for the fundamental reproductive ratio based on the 2D-QTE. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

17 pages, 999 KiB

Open AccessArticle

Classes of Multivalent Spirallike Functions Associated with Symmetric Regions

by Luminiţa-Ioana Cotîrlǎ and Kadhavoor R. Karthikeyan

Symmetry 2022, 14(8), 1598; https://doi.org/10.3390/sym14081598 - 03 Aug 2022

Cited by 2 | Viewed by 1103

Abstract

We define a function to unify the well-known class of Janowski functions with a class of spirallike functions of reciprocal order. We focus on the impact of defined function on various conic regions which are symmetric with respect to the real axis. Further, [...] Read more.

We define a function to unify the well-known class of Janowski functions with a class of spirallike functions of reciprocal order. We focus on the impact of defined function on various conic regions which are symmetric with respect to the real axis. Further, we have defined a new subclass of multivalent functions of complex order subordinate to the extended Janowski function. This work bridges the studies of various subclasses of spirallike functions and extends well-known results. Interesting properties have been obtained for the defined function class. Several consequences of our main results have been pointed out. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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16 pages, 300 KiB

Open AccessArticle

Montgomery Identity and Ostrowski-Type Inequalities for Generalized Quantum Calculus through Convexity and Their Applications

by Humaira Kalsoom, Miguel Vivas-Cortez, Muhammad Zainul Abidin, Muhammad Marwan and Zareen A. Khan

Symmetry 2022, 14(7), 1449; https://doi.org/10.3390/sym14071449 - 15 Jul 2022

Cited by 5 | Viewed by 1295

Abstract

The celebrated Montgomery identity has been studied extensively since it was established. We found a novel version of the Montgomery identity when we were working inside the framework of p- and q-calculus. We acquire a Montgomery identity through a definite [...] Read more.

The celebrated Montgomery identity has been studied extensively since it was established. We found a novel version of the Montgomery identity when we were working inside the framework of p- and q-calculus. We acquire a Montgomery identity through a definite

(p, q)

-integral from these results. Consequently, we establish specific Ostrowski-type

(p, q)

-integral inequalities by using Montgomery identity. In addition to the well-known repercussions, this novel study provides an opportunity to set up new boundaries in the field of comparative literature. The research that is being proposed on the

(p, q)

-integral includes some fascinating results that demonstrate the superiority and applicability of the findings that have been achieved. This highly successful and valuable strategy is anticipated to create a new venue in the contemporary realm of special relativity and quantum theory. These mathematical inequalities and the approaches that are related to them have applications in the areas that deal with symmetry. Additionally, an application to special means is provided in the conclusion. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

13 pages, 319 KiB

Open AccessArticle

New Estimates for Hermite-Hadamard Inequality in Quantum Calculus via (α, m) Convexity

by Peng Xu, Saad Ihsan Butt, Qurat Ul Ain and Hüseyin Budak

Symmetry 2022, 14(7), 1394; https://doi.org/10.3390/sym14071394 - 06 Jul 2022

Cited by 7 | Viewed by 1162

Abstract

This study provokes the existence of quantum Hermite-Hadamard inequalities under the concept of q-integral. We analyse and illustrate a new identity for the differentiable function mappings whose second derivatives in absolute value are

(α, m)

convex. Some basic inequalities [...] Read more.

This study provokes the existence of quantum Hermite-Hadamard inequalities under the concept of q-integral. We analyse and illustrate a new identity for the differentiable function mappings whose second derivatives in absolute value are

(α, m)

convex. Some basic inequalities such as Hölder’s and Power mean have been used to obtain new bounds and it has been determined that the main findings are generalizations of many results that exist in the literature. We make links between our findings and a number of well-known discoveries in the literature. The conclusion in this study unify and generalise previous findings on Hermite-Hadamard inequalities. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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11 pages, 263 KiB

Open AccessArticle

Some New Anderson Type h and q Integral Inequalities in Quantum Calculus

by Munawwar Ali Abbas, Li Chen, Asif R. Khan, Ghulam Muhammad, Bo Sun, Sadaqat Hussain, Javed Hussain and Adeeb Ur Rasool

Symmetry 2022, 14(7), 1294; https://doi.org/10.3390/sym14071294 - 22 Jun 2022

Cited by 2 | Viewed by 1437

Abstract

The calculus in the absence of limits is known as quantum calculus. With a difference operator, it substitutes the classical derivative, which permits dealing with sets of functions that are non-differentiations. The theory of integral inequality in quantum calculus is a field of [...] Read more.

The calculus in the absence of limits is known as quantum calculus. With a difference operator, it substitutes the classical derivative, which permits dealing with sets of functions that are non-differentiations. The theory of integral inequality in quantum calculus is a field of mathematics that has been gaining considerable attention recently. Despite the fact of its application in discrete calculus, it can be applied in fractional calculus as well. In this paper, some new Anderson type q-integral and h-integral inequalities are given using a Feng Qi integral inequality in quantum calculus. These findings are highly beneficial for basic frontier theories, and the techniques offered by technology are extremely useful for those who can stimulate research interest in exploring mathematical applications. Due to the interesting properties in the field of mathematics, integral inequalities have a tied correlation with symmetric convex and convex functions. There exist strong correlations and expansive properties between the different fields of convexity and symmetric function, including probability theory, convex functions, and the geometry of convex functions on convex sets. The main advantage of these essential inequalities is that they can be converted into time-scale calculus. This kind of inevitable inequality can be very helpful in various fields where coordination plays an important role. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

13 pages, 314 KiB

Open AccessArticle

Simpson’s and Newton’s Type Inequalities for (α,m)-Convex Functions via Quantum Calculus

by Jarunee Soontharanon, Muhammad Aamir Ali, Hüseyin Budak, Kamsing Nonlaopon and Zoya Abdullah

Symmetry 2022, 14(4), 736; https://doi.org/10.3390/sym14040736 - 03 Apr 2022

Cited by 17 | Viewed by 1585

Abstract

In this paper, we give the generalized version of the quantum Simpson’s and quantum Newton’s formula type inequalities via quantum differentiable

(α, m)

-convex functions. The main advantage of these new inequalities is that they can be converted into quantum Simpson and [...] Read more.

In this paper, we give the generalized version of the quantum Simpson’s and quantum Newton’s formula type inequalities via quantum differentiable

(α, m)

-convex functions. The main advantage of these new inequalities is that they can be converted into quantum Simpson and quantum Newton for convex functions, Simpson’s type inequalities

(α, m)

-convex function, and Simpson’s type inequalities without proving each separately. These inequalities can be helpful in finding the error bounds of Simpson’s and Newton’s formulas in numerical integration. Analytic inequalities of this type as well as particularly related strategies have applications for various fields where symmetry plays an important role. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

20 pages, 327 KiB

Open AccessArticle

q₁q₂-Ostrowski-Type Integral Inequalities Involving Property of Generalized Higher-Order Strongly n-Polynomial Preinvexity

by Humaira Kalsoom and Miguel Vivas-Cortez

Symmetry 2022, 14(4), 717; https://doi.org/10.3390/sym14040717 - 01 Apr 2022

Cited by 5 | Viewed by 1473

Abstract

Quantum calculus has numerous applications in mathematics. This novel class of functions may be used to produce a variety of conclusions in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities. It can drive additional research in a variety of [...] Read more.

Quantum calculus has numerous applications in mathematics. This novel class of functions may be used to produce a variety of conclusions in convex analysis, special functions, quantum mechanics, related optimization theory, and mathematical inequalities. It can drive additional research in a variety of pure and applied fields. This article’s main objective is to introduce and study a new class of preinvex functions, which is called higher-order generalized strongly n-polynomial preinvex function. We derive a new

q_{1} q_{2}

-integral identity for mixed partial

q_{1} q_{2}

-differentiable functions. Because of the nature of generalized convexity theory, there is a strong link between preinvexity and symmetry. Utilizing this as an auxiliary result, we derive some estimates of upper bound for functions whose mixed partial

q_{1} q_{2}

-differentiable functions are higher-order generalized strongly n-polynomial preinvex functions on co-ordinates. Our results are the generalizations of the results in earlier papers. Quantum inequalities of this type and the techniques used to solve them have applications in a wide range of fields where symmetry is important. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

19 pages, 307 KiB

Open AccessArticle

Multi-Parameter Quantum Integral Identity Involving Raina’s Function and Corresponding q-Integral Inequalities with Applications

by Miguel Vivas-Cortez, Muhammad Uzair Awan, Sadia Talib, Artion Kashuri and Muhammad Aslam Noor

Symmetry 2022, 14(3), 606; https://doi.org/10.3390/sym14030606 - 18 Mar 2022

Cited by 1 | Viewed by 1341

Abstract

Convexity performs its due role in the theoretical field of inequalities according to the nature and conduct of the properties it displays. A correlation connectivity, which is visible between the two variables symmetry and convexity, enhances its importance. In this paper, we derive [...] Read more.

Convexity performs its due role in the theoretical field of inequalities according to the nature and conduct of the properties it displays. A correlation connectivity, which is visible between the two variables symmetry and convexity, enhances its importance. In this paper, we derive a new multi-parameter quantum integral identity involving Raina’s function. Applying this generic identity as an auxiliary result, we establish some new generalized quantum estimates of certain integral inequalities pertaining to the class of

R_{s}

-convex functions. Moreover, we give quantum integral inequalities for the product of

R_{s_{1}}

- and

R_{s_{2}}

-convex functions as well as another quantum result for a function that satisfies a special condition. In order to demonstrate the efficiency of our main results, we offer many important special cases for suitable choices of parameters and finally for

R_{s}

-convex functions that are absolute-value bounded. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

20 pages, 345 KiB

Open AccessArticle

New Simpson’s Type Estimates for Two Newly Defined Quantum Integrals

by Muhammad Raees, Matloob Anwar, Miguel Vivas-Cortez, Artion Kashuri, Muhammad Samraiz and Gauhar Rahman

Symmetry 2022, 14(3), 548; https://doi.org/10.3390/sym14030548 - 08 Mar 2022

Cited by 2 | Viewed by 1491

Abstract

In this paper, we give some correct quantum type Simpson’s inequalities via the application of q-Hölder’s inequality. The inequalities of this study are compatible with famous Simpson’s

1 / 8

and

3 / 8

quadrature rules for four and six panels, respectively. [...] Read more.

In this paper, we give some correct quantum type Simpson’s inequalities via the application of q-Hölder’s inequality. The inequalities of this study are compatible with famous Simpson’s

1 / 8

and

3 / 8

quadrature rules for four and six panels, respectively. Several special cases from our results are discussed in detail. A counter example is presented to explain the limitation of Hölder’s inequality in the quantum framework. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

15 pages, 315 KiB

Open AccessArticle

Hadamard-Type Inequalities for Generalized Integral Operators Containing Special Functions

by Chahnyong Jung, Ghulam Farid, Muhammad Yussouf and Kamsing Nonlaopon

Symmetry 2022, 14(3), 492; https://doi.org/10.3390/sym14030492 - 28 Feb 2022

Viewed by 1475

Abstract

Convex functions are studied very frequently by means of the Hadamard inequality. A symmetric function leads to the generalization of the Hadamard inequality; the Fejér–Hadamard inequality is one of the generalizations of the Hadamard inequality that holds for convex functions defined on a [...] Read more.

Convex functions are studied very frequently by means of the Hadamard inequality. A symmetric function leads to the generalization of the Hadamard inequality; the Fejér–Hadamard inequality is one of the generalizations of the Hadamard inequality that holds for convex functions defined on a finite interval along with functions which have symmetry about the midpoint of that finite interval. Lately, integral inequalities for convex functions have been extensively generalized by fractional integral operators. In this paper, inequalities of Hadamard type are generalized by using exponentially (

α

, h-m)-p-convex functions and an operator containing an extended generalized Mittag-Leffler function. The obtained results are also connected with several well-known Hadamard-type inequalities. Full article

(This article belongs to the Special Issue Symmetry in Quantum Calculus)

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