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Fractal Fract
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Fractional Integral Inequalities and Applications, 2nd Edition
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Fractional Integral Inequalities and Applications, 2nd Edition

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (30 June 2024) | Viewed by 18009

Special Issue Editors

Dr. Artion Kashuri

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora 9401, Albania
Interests: mathematical inequalities; special functions; approximation theory; fractional calculus; applied mathematics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hari Mohan Srivastava

grade E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modelling and optimization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The beauty and amazing theory of inequalities represents a long-standing topic in many different mathematical areas and remains an attractive research area with many interesting applications in fractional calculus, quantum calculus, operator theory, numerical analysis, operator equations, network theory and quantum information theory. Research in these subjects has been very lively recently, and the interplay between individual areas has enriched them all.

The numerical integration and the numerical estimations of definite integrals are vital pieces of applied sciences. Simpson's rules are momentous among the numerical techniques.

This Special Issue brings together original research papers in all areas of mathematics and its numerous applications that are concerned with inequalities or their basic role. The research results presented are related to the improvement, extensions and generalizations of classical and recent inequalities, and highlight their applications in functional analysis, nonlinear functional analysis, multivariate analysis, quantum calculus, statistics, probability and other fields.

Please note that all submitted papers should be within the scope of the journal.

Dr. Artion Kashuri
Prof. Dr. Hari Mohan Srivastava
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional integral inequalities
  • generalized convexity
  • numerical estimations
  • quantum calculus
  • multivariate analysis
  • means
  • operator theory
  • approximation theory

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Related Special Issue

Published Papers (14 papers)

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Research

24 pages, 455 KiB  
Article
On Newton–Cotes Formula-Type Inequalities for Multiplicative Generalized Convex Functions via Riemann–Liouville Fractional Integrals with Applications to Quadrature Formulas and Computational Analysis
by Abdul Mateen, Serap Özcan, Zhiyue Zhang and Bandar Bin-Mohsin
Fractal Fract. 2024, 8(9), 541; https://doi.org/10.3390/fractalfract8090541 - 18 Sep 2024
Viewed by 509
Abstract
In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex [...] Read more.
In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
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20 pages, 332 KiB  
Article
β–Ulam–Hyers Stability and Existence of Solutions for Non-Instantaneous Impulsive Fractional Integral Equations
by Wei-Shih Du, Michal Fečkan, Marko Kostić and Daniel Velinov
Fractal Fract. 2024, 8(8), 469; https://doi.org/10.3390/fractalfract8080469 - 12 Aug 2024
Viewed by 966
Abstract
In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within [...] Read more.
In this paper, we investigate a class of non-instantaneous impulsive fractional integral equations. Utilizing the Banach contraction mapping principle, we establish the existence and uniqueness of solutions for the considered problem. Additionally, employing Schauder’s fixed-point theorem, we demonstrate the existence of solutions within the framework of β-Banach spaces. Moreover, we examine the β–Ulam–Hyers stability of the solutions, providing insights into the stability behavior under small perturbations. An illustrative example is presented to demonstrate the practical applicability and effectiveness of the theoretical results obtained. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
15 pages, 320 KiB  
Article
An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions
by Asifa Tassaddiq, Rekha Srivastava, Rabab Alharbi, Ruhaila Md Kasmani and Sania Qureshi
Fractal Fract. 2024, 8(8), 438; https://doi.org/10.3390/fractalfract8080438 - 25 Jul 2024
Viewed by 827
Abstract
This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in the finite interval a≤t≤x [...] Read more.
This research aims to develop generalized fractional integral inequalities by utilizing multiple Erdélyi–Kober (E–K) fractional integral operators. Using a set of j, with (j∈N) positively continuous and decaying functions in the finite interval a≤t≤x, the Fox-H function is involved in establishing new and novel fractional integral inequalities. Since the Fox-H function is the most general special function, the obtained inequalities are therefore sufficiently widespread and significant in comparison to the current literature to yield novel and unique results. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
26 pages, 565 KiB  
Article
Fractional Hermite–Hadamard–Mercer-Type Inequalities for Interval-Valued Convex Stochastic Processes with Center-Radius Order and Their Related Applications in Entropy and Information Theory
by Ahsan Fareed Shah, Serap Özcan, Miguel Vivas-Cortez, Muhammad Shoaib Saleem and Artion Kashuri
Fractal Fract. 2024, 8(7), 408; https://doi.org/10.3390/fractalfract8070408 - 11 Jul 2024
Viewed by 1182
Abstract
We propose a new definition of the γ-convex stochastic processes (CSP) using center and radius (CR) order with the notion of interval valued functions (C.RI.V). By utilizing this definition [...] Read more.
We propose a new definition of the γ-convex stochastic processes (CSP) using center and radius (CR) order with the notion of interval valued functions (C.RI.V). By utilizing this definition and Mean-Square Fractional Integrals, we generalize fractional Hermite–Hadamard–Mercer-type inclusions for generalized C.RI.V versions of convex, tgs-convex, P-convex, exponential-type convex, Godunova–Levin convex, s-convex, Godunova–Levin s-convex, h-convex, n-polynomial convex, and fractional n-polynomial (CSP). Also, our work uses interesting examples of C.RI.V(CSP) with Python-programmed graphs to validate our findings using an extension of Mercer’s inclusions with applications related to entropy and information theory. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
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12 pages, 326 KiB  
Article
Fuzzy Differential Subordination for Classes of Admissible Functions Defined by a Class of Operators
by Ekram E. Ali, Miguel Vivas-Cortez and Rabha M. El-Ashwah
Fractal Fract. 2024, 8(7), 405; https://doi.org/10.3390/fractalfract8070405 - 11 Jul 2024
Cited by 3 | Viewed by 730
Abstract
This paper’s findings are related to geometric function theory (GFT). We employ one of the most recent methods in this area, the fuzzy admissible functions methodology, which is based on fuzzy differential subordination, to produce them. To do this, the relevant fuzzy admissible [...] Read more.
This paper’s findings are related to geometric function theory (GFT). We employ one of the most recent methods in this area, the fuzzy admissible functions methodology, which is based on fuzzy differential subordination, to produce them. To do this, the relevant fuzzy admissible function classes must first be defined. This work deals with fuzzy differential subordinations, ideas borrowed from fuzzy set theory and applied to complex analysis. This work examines the characteristics of analytic functions and presents a class of operators in the open unit disk Jη,ςκ(a,e,x) for ς>−1,η>0, such that a,e∈R,(e−a)≥0,a>−x. The fuzzy differential subordination results are obtained using (GFT) concepts outside the field of complex analysis because of the operator’s compositional structure, and some relevant classes of admissible functions are studied by utilizing fuzzy differential subordination. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
19 pages, 456 KiB  
Article
Some New Approaches to Fractional Euler–Maclaurin-Type Inequalities via Various Function Classes
by Mehmet Gümüş, Fatih Hezenci and Hüseyin Budak
Fractal Fract. 2024, 8(7), 372; https://doi.org/10.3390/fractalfract8070372 - 26 Jun 2024
Viewed by 1325
Abstract
This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions [...] Read more.
This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions by fractional integrals. Some fractional Euler–Maclaurin-type inequalities are established for Lipschitzian functions. Finally, several Euler–Maclaurin-type inequalities are constructed by fractional integrals of bounded variation. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
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13 pages, 325 KiB  
Article
Fuzzy Subordination Results for Meromorphic Functions Connected with a Linear Operator
by Ekram E. Ali, Miguel Vivas-Cortez, Rabha M. El-Ashwah and Abeer M. Albalahi
Fractal Fract. 2024, 8(6), 308; https://doi.org/10.3390/fractalfract8060308 - 23 May 2024
Cited by 1 | Viewed by 876
Abstract
The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this [...] Read more.
The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work’s assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
16 pages, 311 KiB  
Article
Novel Estimations of Hadamard-Type Integral Inequalities for Raina’s Fractional Operators
by Merve Coşkun, Çetin Yildiz and Luminiţa-Ioana Cotîrlă
Fractal Fract. 2024, 8(5), 302; https://doi.org/10.3390/fractalfract8050302 - 20 May 2024
Viewed by 1071
Abstract
In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using [...] Read more.
In the present paper, utilizing a wide class of fractional integral operators (namely the Raina fractional operator), we develop novel fractional integral inequalities of the Hermite–Hadamard type. With the help of the well-known Riemann–Liouville fractional operators, s-type convex functions are derived using the important results. We also note that some of the conclusions of this study are more reasonable than those found under certain specific conditions, e.g., s=1, λ=α, σ(0)=1, and w=0. In conclusion, the methodology described in this article is expected to stimulate further research in this area. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
16 pages, 310 KiB  
Article
Error Bounds for Fractional Integral Inequalities with Applications
by Nouf Abdulrahman Alqahtani, Shahid Qaisar, Arslan Munir, Muhammad Naeem and Hüseyin Budak
Fractal Fract. 2024, 8(4), 208; https://doi.org/10.3390/fractalfract8040208 - 2 Apr 2024
Cited by 2 | Viewed by 1113
Abstract
Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson’s type estimates by employing a differentiable function. Furthermore, a novel class of [...] Read more.
Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson’s type estimates by employing a differentiable function. Furthermore, a novel class of fractional integral related to prominent fractional operator (Caputo–Fabrizio) for differentiable convex functions of first order is proven. Then, taking this equality into account as an auxiliary result, some new estimation of the Hermite–Hadamard and Simpson’s type inequalities as generalization is presented. Moreover, few inequalities for concave function are obtained as well. It is observed that newly established outcomes are the extension of comparable inequalities existing in the literature. Additionally, we discuss the applications to special means, matrix inequalities, and the q-digamma function. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
15 pages, 355 KiB  
Article
New Inequalities Using Multiple Erdélyi–Kober Fractional Integral Operators
by Asifa Tassaddiq, Rekha Srivastava, Rabab Alharbi, Ruhaila Md Kasmani and Sania Qureshi
Fractal Fract. 2024, 8(4), 180; https://doi.org/10.3390/fractalfract8040180 - 22 Mar 2024
Cited by 6 | Viewed by 1328
Abstract
The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new [...] Read more.
The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
15 pages, 340 KiB  
Article
On q-Hermite–Hadamard Type Inequalities via s-Convexity and (α,m)-Convexity
by Loredana Ciurdariu and Eugenia Grecu
Fractal Fract. 2024, 8(1), 12; https://doi.org/10.3390/fractalfract8010012 - 22 Dec 2023
Cited by 1 | Viewed by 1229
Abstract
The purpose of the paper is to present new q-parametrized Hermite–Hadamard-like type integral inequalities for functions whose third quantum derivatives in absolute values are s-convex and (α,m)-convex, respectively. Two new q-integral identities are presented for [...] Read more.
The purpose of the paper is to present new q-parametrized Hermite–Hadamard-like type integral inequalities for functions whose third quantum derivatives in absolute values are s-convex and (α,m)-convex, respectively. Two new q-integral identities are presented for three time q-differentiable functions. These lemmas are used like basic elements in our proofs, along with several important tools like q-power mean inequality, and q-Holder’s inequality. In a special case, a non-trivial example is considered for a specific parameter and this case illustrates the investigated results. We make links between these findings and several previous discoveries from the literature. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
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12 pages, 305 KiB  
Article
On the Generalization of Ostrowski-Type Integral Inequalities via Fractional Integral Operators with Application to Error Bounds
by Gauhar Rahman, Miguel Vivas-Cortez, Çetin Yildiz, Muhammad Samraiz, Shahid Mubeen and Mansour F. Yassen
Fractal Fract. 2023, 7(9), 683; https://doi.org/10.3390/fractalfract7090683 - 14 Sep 2023
Viewed by 2021
Abstract
The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The Ostrowski’s type inequality is frequently used to investigate errors in numerical quadrature rules and computations. In this work, Ostrowski-type inequality is demonstrated using the generalized fractional integral [...] Read more.
The Ostrowski inequality expresses bounds on the deviation of a function from its integral mean. The Ostrowski’s type inequality is frequently used to investigate errors in numerical quadrature rules and computations. In this work, Ostrowski-type inequality is demonstrated using the generalized fractional integral operators. From an application perspective, we present the bounds of the fractional Hadamard inequalities. The results that are being presented involve a number of fractional inequalities that are already known and have been published. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
19 pages, 390 KiB  
Article
New Estimates on Hermite–Hadamard Type Inequalities via Generalized Tempered Fractional Integrals for Convex Functions with Applications
by Artion Kashuri, Yahya Almalki, Ali M. Mahnashi and Soubhagya Kumar Sahoo
Fractal Fract. 2023, 7(8), 579; https://doi.org/10.3390/fractalfract7080579 - 27 Jul 2023
Cited by 2 | Viewed by 1414
Abstract
This paper presents a novel approach by introducing a set of operators known as the left and right generalized tempered fractional integral operators. These operators are utilized to establish new Hermite–Hadamard inequalities for convex functions as well as the multiplication of two convex [...] Read more.
This paper presents a novel approach by introducing a set of operators known as the left and right generalized tempered fractional integral operators. These operators are utilized to establish new Hermite–Hadamard inequalities for convex functions as well as the multiplication of two convex functions. Additionally, this paper gives two useful identities involving the generalized tempered fractional integral operator for differentiable functions. By leveraging these identities, our results consist of integral inequalities of the Hermite–Hadamard type, which are specifically designed to accommodate convex functions. Furthermore, this study encompasses the identification of several special cases and the recovery of specific known results through comprehensive research. Lastly, this paper offers a range of applications in areas such as matrices, modified Bessel functions and q-digamma functions. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
13 pages, 317 KiB  
Article
On Some New Maclaurin’s Type Inequalities for Convex Functions in q-Calculus
by Thanin Sitthiwirattham, Muhammad Aamir Ali and Hüseyin Budak
Fractal Fract. 2023, 7(8), 572; https://doi.org/10.3390/fractalfract7080572 - 25 Jul 2023
Cited by 4 | Viewed by 843
Abstract
This work establishes some new inequalities to find error bounds for Maclaurin’s formulas in the framework of q-calculus. For this, we first prove an integral identity involving q-integral and q-derivative. Then, we use this new identity to prove some q [...] Read more.
This work establishes some new inequalities to find error bounds for Maclaurin’s formulas in the framework of q-calculus. For this, we first prove an integral identity involving q-integral and q-derivative. Then, we use this new identity to prove some q-integral inequalities for q-differentiable convex functions. The inequalities proved here are very important in the literature because, with their help, we can find error bounds for Maclaurin’s formula in both q and classical calculus. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)
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