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Fractal Fract
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Fractional Integral Inequalities and Applications, 3rd Edition
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Fractional Integral Inequalities and Applications, 3rd 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: 28 February 2026 | Viewed by 7389

Special Issue Editors

Dr. Artion Kashuri

E-Mail Website
Guest Editor
Department of Mathematical Engineering, Polytechnic University of Tirana, 1001 Tirana, 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 theory of inequalities represents a long-standing topic in many areas of mathematics and remains an attractive research area, with applications spanning fractional calculus, quantum calculus, operator theory, numerical analysis, operator equations, network theory and quantum information theory. In recent years, this area of research has garnered significant attention, and the interplay between individual aspects of this theory has enriched all of these domains.

The numerical integration and estimation of definite integrals is vital in applied sciences. Among the numerical techniques, Simpson's rules are particularly notable.

This Special Issue aims to collect original research papers that address all areas of mathematics and the numerous applications concerned with inequalities or their basic role. The research results presented should be related to the improvement, extension and generalization of classical and recent inequalities, and highlight their application 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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Published Papers (9 papers)

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Research

27 pages, 556 KB  
Article
Refined Error Estimates for Milne–Mercer-Type Inequalities for Three-Times-Differentiable Functions with Error Analysis and Their Applications
by Arslan Munir, Shumin Li, Hüseyin Budak, Artion Kashuri and Loredana Ciurdariu
Fractal Fract. 2025, 9(9), 606; https://doi.org/10.3390/fractalfract9090606 - 18 Sep 2025
Viewed by 140
Abstract
In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a [...] Read more.
In this study, we examine the error bounds related to Milne-type inequalities and a widely recognized Newton–Cotes method, originally developed for three-times-differentiable convex functions within the context of Jensen–Mercer inequalities. Expanding on this foundation, we explore Milne–Mercer-type inequalities and their application to a more refined class of three-times-differentiable s-convex functions. This work introduces a new identity involving such functions and Jensen–Mercer inequalities, which is then used to improve the error bounds for Milne-type inequalities in both Jensen–Mercer and classical calculus frameworks. Our research highlights the importance of convexity principles and incorporates the power mean inequality to derive novel inequalities. Furthermore, we provide a new lemma using Caputo–Fabrizio fractional integral operators and apply it to derive several results of Milne–Mercer-type inequalities pertaining to (α,m)-convex functions. Additionally, we extend our findings to various classes of functions, including bounded and Lipschitzian functions, and explore their applications to special means, the q-digamma function, the modified Bessel function, and quadrature formulas. We also provide clear mathematical examples to demonstrate the effectiveness of the newly derived bounds for Milne–Mercer-type inequalities. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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26 pages, 717 KB  
Article
Evolutionary Approach to Inequalities of Hermite–Hadamard–Mercer Type for Generalized Wright’s Functions Associated with Computational Evaluation and Their Applications
by Talib Hussain, Loredana Ciurdariu and Eugenia Grecu
Fractal Fract. 2025, 9(9), 593; https://doi.org/10.3390/fractalfract9090593 - 10 Sep 2025
Viewed by 295
Abstract
The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general [...] Read more.
The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general functions, referred to as Raina’s functions in the scientific literature. The main goal of our progressive study is to use Raina’s Fractional Integrals to derive two useful lemmas for second-differentiable functions. Using the derived lemmas, we proved a large number of fractional integral inequalities related to trapezoidal and midpoint-type inequalities where those that are twice differentiable in absolute values are convex. Some of these results also generalize findings from previous research. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, the results are well-supported by numerous examples, including graphical representations and numerical tables, which collectively highlight their accuracy and computational significance. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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23 pages, 323 KB  
Article
Analytical Investigations into Multilinear Fractional Rough Hardy Operators Within Morrey–Herz Spaces Characterized by Variable Exponents
by Muhammad Asim and Ghada AlNemer
Fractal Fract. 2025, 9(9), 573; https://doi.org/10.3390/fractalfract9090573 - 30 Aug 2025
Viewed by 357
Abstract
In this scholarly discourse, a rigorous examination is conducted on the boundedness properties of multilinear fractional rough Hardy operators within the structural framework of variable exponent Morrey–Herz spaces. Furthermore, analogous quantitative estimates are meticulously derived for their corresponding commutators, contingent upon the assumption [...] Read more.
In this scholarly discourse, a rigorous examination is conducted on the boundedness properties of multilinear fractional rough Hardy operators within the structural framework of variable exponent Morrey–Herz spaces. Furthermore, analogous quantitative estimates are meticulously derived for their corresponding commutators, contingent upon the assumption that the governing symbol functions belong to the space of bounded mean oscillation (BMO) with variable exponents. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
27 pages, 3680 KB  
Article
Fuzzy Convexity Under cr-Order with Control Operator and Fractional Inequalities
by Qi Liu, Muhammad Zakria Javed, Muhammad Uzair Awan, Loredana Ciurdariu and Badr S. Alkahtani
Fractal Fract. 2025, 9(6), 391; https://doi.org/10.3390/fractalfract9060391 - 18 Jun 2025
Viewed by 370
Abstract
This study is organized to introduce the concept of center–radius (cr)-ordered fuzzy number-valued convex mappings. Based on this class of mappings, we have initiated the idea of fuzzy number-valued extended cr-ℏ convex mappings incorporating control mapping [...] Read more.
This study is organized to introduce the concept of center–radius (cr)-ordered fuzzy number-valued convex mappings. Based on this class of mappings, we have initiated the idea of fuzzy number-valued extended cr-ℏ convex mappings incorporating control mapping ℏ. Furthermore, several potential new classes of convexity will be provided to discuss its generic nature. Also, some essential properties, criteria, and detailed characterizations through Jensen’s and Hermite–Hadamard-like inequalities are provided, incorporating Riemann–Liouville fractional operators, which are defined by ρ-level mappings. To validate the proposed fractional bounds through simulations, we consider both triangular and trapezoidal fuzzy numbers. Our results are based on totally ordered fuzzy-valued mappings, which are new and generic. The under-consideration class also includes a blend of new classes of convexity, which are controlled by non-negative mapping ℏ. In previous studies, the researchers have focused on different partially ordered relations. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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28 pages, 603 KB  
Article
New Results on Majorized Discrete Jensen–Mercer Inequality for Raina Fractional Operators
by Çetin Yildiz, Tevfik İşleyen and Luminiţa-Ioana Cotîrlă
Fractal Fract. 2025, 9(6), 343; https://doi.org/10.3390/fractalfract9060343 - 26 May 2025
Viewed by 385
Abstract
As the most important inequality, the Hermite–Hadamard–Mercer inequality has attracted the interest of numerous additional mathematicians. Numerous findings on this inequality have been developed in recent years. So, in this paper, we demonstrate novel Hermite–Hadamard–Mercer inequalities using Raina fractional operators and the majorization [...] Read more.
As the most important inequality, the Hermite–Hadamard–Mercer inequality has attracted the interest of numerous additional mathematicians. Numerous findings on this inequality have been developed in recent years. So, in this paper, we demonstrate novel Hermite–Hadamard–Mercer inequalities using Raina fractional operators and the majorization concept. Furthermore, additional identities are discovered, and two new lemmas of this type are proved. A summary of several known results is also provided, along with a thorough derivation of some exceptional cases. We also note that some of the outcomes in this study are more acceptable than others under certain exceptional instances, such as setting n=2, w=0, σ(0)=1, and λ=1 or λ=α. Lastly, the method described in this publication is thought to stimulate further research in this area. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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23 pages, 358 KB  
Article
New Approaches to Fractal–Fractional Bullen’s Inequalities Through Generalized Convexity
by Wedad Saleh, Hamid Boulares, Abdelkader Moumen, Hussien Albala and Badreddine Meftah
Fractal Fract. 2025, 9(1), 25; https://doi.org/10.3390/fractalfract9010025 - 3 Jan 2025
Cited by 5 | Viewed by 1381
Abstract
This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals [...] Read more.
This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals and fractal calculus, but also offering a refinement of the well-known Bullen-type inequality. We further explore the connections between generalized convexity and fractal–fractional integrals, showing how the concept of generalized convexity enables the establishment of error bounds for fractal–fractional integrals involving lower-order derivatives, with an emphasis on their applications in various fields. The findings expand the current understanding of fractal–fractional inequalities and offer new insights into the use of local fractional derivatives for analyzing functions with fractional-order properties. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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21 pages, 483 KB  
Article
New Inequalities for GA–h Convex Functions via Generalized Fractional Integral Operators with Applications to Entropy and Mean Inequalities
by Asfand Fahad, Zammad Ali, Shigeru Furuichi, Saad Ihsan Butt, Ayesha and Yuanheng Wang
Fractal Fract. 2024, 8(12), 728; https://doi.org/10.3390/fractalfract8120728 - 12 Dec 2024
Cited by 2 | Viewed by 988
Abstract
We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional [...] Read more.
We prove the inequalities of the weighted Hermite–Hadamard type the and Hermite–Hadamard–Mercer type for an extremely rich class of geometrically arithmetically-h-convex functions (GA-h-CFs) via generalized Hadamard–Fractional integral operators (HFIOs). The two generalized fractional integral operators (FIOs) are Hadamard proportional fractional integral operators (HPFIOs) and Hadamard k-fractional integral operators (HKFIOs). Moreover, we also present the results for subclasses of GA-h-CFs and show that the inequalities proved in this paper unify the results from the recent related literature. Furthermore, we compare the two generalizations in view of the fractional operator parameters that contribute to the generalizations of the results and assess the better approximation via graphical tools. Finally, we present applications of the new inequalities via HPFIOs and HKFIOs by establishing interpolation relations between arithmetic mean and geometric mean and by proving the new upper bounds for the Tsallis relative operator entropy. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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17 pages, 335 KB  
Article
Significant Study of Fuzzy Fractional Inequalities with Generalized Operators and Applications
by Rana Safdar Ali, Humira Sif, Gauhar Rehman, Ahmad Aloqaily and Nabil Mlaiki
Fractal Fract. 2024, 8(12), 690; https://doi.org/10.3390/fractalfract8120690 - 24 Nov 2024
Cited by 1 | Viewed by 722
Abstract
There are many techniques for the extension and generalization of fractional theories, one of which improves fractional operators by means of their kernels. This paper is devoted to the most general concept of interval-valued functions, studying fractional integral operators for interval-valued functions, along [...] Read more.
There are many techniques for the extension and generalization of fractional theories, one of which improves fractional operators by means of their kernels. This paper is devoted to the most general concept of interval-valued functions, studying fractional integral operators for interval-valued functions, along with the multi-variate extension of the Bessel–Maitland function, which acts as kernel. We discuss the behavior of Hermite–Hadamard Fejér (HHF)-type inequalities by using the convex fuzzy interval-valued function (C-FIVF) with generalized fuzzy fractional operators. Also, we obtain some refinements of Hermite–Hadamard(H-H)-type inequalities via convex fuzzy interval-valued functions (C-FIVFs). Our results extend and generalize existing findings from the literature. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
23 pages, 504 KB  
Article
Fractional Reverse Inequalities Involving Generic Interval-Valued Convex Functions and Applications
by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Badreddine Meftah and Artion Kashuri
Fractal Fract. 2024, 8(10), 587; https://doi.org/10.3390/fractalfract8100587 - 3 Oct 2024
Cited by 7 | Viewed by 1463
Abstract
The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (H-H)-like inequalities using fractional calculus [...] Read more.
The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (H-H)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of I.V-(⋏,ℏ) generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and I.V-(⋏,ℏ) convexity to derive new improvements of the H-H- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for I.V R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)
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