Fractional Integral Inequalities and Applications

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 (31 January 2023) | Viewed by 27380

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Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora 9401, Albania
Interests: mathematical inequalities; special functions; approximation theory; fractional calculus; applied mathematics
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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 modeling and optimization
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Special Issue Information

Dear Colleagues,

The amazing theory of inequalities is a long-standing topic in many different mathematical areas. It remains an attractive research subject, with many interesting applications in fractional calculus, quantum calculus, operator theory, numerical analysis, operator equations, network theory and quantum information theory. This is currently a very active research area, which has been enriched by interplay between individual areas.

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

This Special Issue will collate 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 relevant 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 (16 papers)

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Research

18 pages, 343 KiB  
Article
On Fractional Integral Inequalities of Riemann Type for Composite Convex Functions and Applications
by Miguel Vivas-Cortez, Muzammil Mukhtar, Iram Shabbir, Muhammad Samraiz and Muhammad Yaqoob
Fractal Fract. 2023, 7(5), 345; https://doi.org/10.3390/fractalfract7050345 - 22 Apr 2023
Cited by 2 | Viewed by 1307
Abstract
In this study, we apply fractional calculus on certain convex functions and derive many novel mean-type inequalities by employing fractional calculus and convexity theory. In order to investigate fractional mean inequalities, we first build an identity in this study. Then, with its help, [...] Read more.
In this study, we apply fractional calculus on certain convex functions and derive many novel mean-type inequalities by employing fractional calculus and convexity theory. In order to investigate fractional mean inequalities, we first build an identity in this study. Then, with its help, we derive many mean-type inequalities and estimate the error of HH inequality using a generalized version of RL-fractional integrals and certain classes of convex functions. The results obtained are validated by taking specific functions. Many mean-type inequalities that exist in the literature are generalized by the main results of this study. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
12 pages, 296 KiB  
Article
New Hadamard Type Inequalities for Modified h-Convex Functions
by Daniel Breaz, Çetin Yildiz, Luminiţa-Ioana Cotîrlă, Gauhar Rahman and Büşra Yergöz
Fractal Fract. 2023, 7(3), 216; https://doi.org/10.3390/fractalfract7030216 - 25 Feb 2023
Cited by 9 | Viewed by 1261
Abstract
In this article, we demonstrated various Hermite–Hadamard and Fejér type inequalities for modified h-convex functions. We showed several inequalities for the products of two modified h-convex functions. New identities related to inequalities in various forms are also established for different values [...] Read more.
In this article, we demonstrated various Hermite–Hadamard and Fejér type inequalities for modified h-convex functions. We showed several inequalities for the products of two modified h-convex functions. New identities related to inequalities in various forms are also established for different values of the h(φt) function. We believe that the approach presented in this paper will inspire more research in this area. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
15 pages, 858 KiB  
Article
On Ostrowski–Mercer’s Type Fractional Inequalities for Convex Functions and Applications
by Soubhagya Kumar Sahoo, Artion Kashuri, Munirah Aljuaid, Soumyarani Mishra and Manuel De La Sen
Fractal Fract. 2023, 7(3), 215; https://doi.org/10.3390/fractalfract7030215 - 25 Feb 2023
Cited by 6 | Viewed by 1474
Abstract
This research focuses on the Ostrowski–Mercer inequalities, which are presented as variants of Jensen’s inequality for differentiable convex functions. The main findings were effectively composed of convex functions and their properties. The results were directed by Riemann–Liouville fractional integral operators. Furthermore, using special [...] Read more.
This research focuses on the Ostrowski–Mercer inequalities, which are presented as variants of Jensen’s inequality for differentiable convex functions. The main findings were effectively composed of convex functions and their properties. The results were directed by Riemann–Liouville fractional integral operators. Furthermore, using special means, q-digamma functions and modified Bessel functions, some applications of the acquired results were obtained. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
15 pages, 324 KiB  
Article
Some New Fractal Milne-Type Integral Inequalities via Generalized Convexity with Applications
by Badreddine Meftah, Abdelghani Lakhdari, Wedad Saleh and Adem Kiliçman
Fractal Fract. 2023, 7(2), 166; https://doi.org/10.3390/fractalfract7020166 - 7 Feb 2023
Cited by 12 | Viewed by 1237
Abstract
This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving first-order generalized derivatives. Finally, as applications, some [...] Read more.
This study aims to construct some new Milne-type integral inequalities for functions whose modulus of the local fractional derivatives is convex on the fractal set. To that end, we develop a novel generalized integral identity involving first-order generalized derivatives. Finally, as applications, some error estimates for the Milne-type quadrature formula and new inequalities for the generalized arithmetic and p-Logarithmic means are derived. This paper’s findings represent a significant improvement over previously published results. The paper’s ideas and formidable tools may inspire and motivate further research in this worthy and fascinating field. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
17 pages, 340 KiB  
Article
Corrected Dual-Simpson-Type Inequalities for Differentiable Generalized Convex Functions on Fractal Set
by Abdelghani Lakhdari, Wedad Saleh, Badreddine Meftah and Akhlad Iqbal
Fractal Fract. 2022, 6(12), 710; https://doi.org/10.3390/fractalfract6120710 - 29 Nov 2022
Cited by 12 | Viewed by 1120
Abstract
The present paper provides several corrected dual-Simpson-type inequalities for functions whose local fractional derivatives are generalized convex. To that end, we derive a new local fractional integral identity as an auxiliary result. Using this new identity along with generalized Hölder’s inequality and generalized [...] Read more.
The present paper provides several corrected dual-Simpson-type inequalities for functions whose local fractional derivatives are generalized convex. To that end, we derive a new local fractional integral identity as an auxiliary result. Using this new identity along with generalized Hölder’s inequality and generalized power mean inequality, we establish some new variants of fractal corrected dual-Simpson-type integral inequalities. Furthermore, some applications for error estimates of quadrature formulas as well as some special means involving arithmetic and p-logarithmic mean are offered to demonstrate the efficacy of our findings. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
15 pages, 2402 KiB  
Article
A Stochastic Bayesian Neural Network for the Mosquito Dispersal Mathematical System
by Suthep Suantai, Zulqurnain Sabir, Muhammad Asif Zahoor Raja and Watcharaporn Cholamjiak
Fractal Fract. 2022, 6(10), 604; https://doi.org/10.3390/fractalfract6100604 - 16 Oct 2022
Cited by 2 | Viewed by 1576
Abstract
The objective of this study is to examine numerical evaluations of the mosquito dispersal mathematical system (MDMS) in a heterogeneous atmosphere through artificial intelligence (AI) techniques via Bayesian regularization neural networks (BSR-NNs). The MDMS is constructed with six classes, i.e., eggs, larvae, pupae, [...] Read more.
The objective of this study is to examine numerical evaluations of the mosquito dispersal mathematical system (MDMS) in a heterogeneous atmosphere through artificial intelligence (AI) techniques via Bayesian regularization neural networks (BSR-NNs). The MDMS is constructed with six classes, i.e., eggs, larvae, pupae, host, resting mosquito, and ovipositional site densities-based ODEs system. The computing BSR-NNs scheme is applied for three different performances using the data of training, testing and verification, which is divided as 75%, 15%, 10% with twelve hidden neurons. The result comparisons are provided to check the authenticity of the designed AI method portrayed by the BSR-NNs. The AI based BSR-NNs procedure is executed to reduce the mean square error (MSE) for the MDMS. The achieved performances are also presented to validate the efficiency of BSR-NNs scheme using the process of MSE, correlation, error histograms and regression. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
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13 pages, 308 KiB  
Article
The Grüss-Type and Some Other Related Inequalities via Fractional Integral with Respect to Multivariate Mittag-Leffler Function
by Yabin Shao, Gauhar Rahman, Yasser Elmasry, Muhammad Samraiz, Artion Kashuri and Kamsing Nonlaopon
Fractal Fract. 2022, 6(10), 546; https://doi.org/10.3390/fractalfract6100546 - 27 Sep 2022
Cited by 2 | Viewed by 1312
Abstract
In the recent era of research, the field of integral inequalities has earned more recognition due to its wide applications in diverse domains. The researchers have widely studied the integral inequalities by utilizing different approaches. In this present article, we aim to develop [...] Read more.
In the recent era of research, the field of integral inequalities has earned more recognition due to its wide applications in diverse domains. The researchers have widely studied the integral inequalities by utilizing different approaches. In this present article, we aim to develop a variety of certain new inequalities using the generalized fractional integral in the sense of multivariate Mittag-Leffler (M-L) functions, including Grüss-type and some other related inequalities. Also, we use the relationship between the Riemann-Liouville integral, the Prabhakar integral, and the generalized fractional integral to deduce specific findings. Moreover, we support our findings by presenting examples and corollaries. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
13 pages, 345 KiB  
Article
Further Midpoint Inequalities via Generalized Fractional Operators in Riemann–Liouville Sense
by Abd-Allah Hyder, Hüseyin Budak and Areej A. Almoneef
Fractal Fract. 2022, 6(9), 496; https://doi.org/10.3390/fractalfract6090496 - 5 Sep 2022
Cited by 8 | Viewed by 1230
Abstract
In this study, new midpoint-type inequalities are given through recently generalized Riemann–Liouville fractional integrals. Foremost, we present an identity for a class of differentiable functions including the proposed fractional integrals. Then, several midpoint-type inequalities containing generalized Riemann–Liouville fractional integrals are proved by employing [...] Read more.
In this study, new midpoint-type inequalities are given through recently generalized Riemann–Liouville fractional integrals. Foremost, we present an identity for a class of differentiable functions including the proposed fractional integrals. Then, several midpoint-type inequalities containing generalized Riemann–Liouville fractional integrals are proved by employing the features of convex and concave functions. Furthermore, all obtained results in this study can be compared to previously published results. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
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10 pages, 287 KiB  
Article
Certain Weighted Fractional Inequalities via the Caputo–Fabrizio Approach
by Vaijanath L. Chinchane, Asha B. Nale, Satish K. Panchal and Christophe Chesneau
Fractal Fract. 2022, 6(9), 495; https://doi.org/10.3390/fractalfract6090495 - 5 Sep 2022
Cited by 2 | Viewed by 1238
Abstract
The Caputo–Fabrizio fractional integral operator is one of the important notions of fractional calculus. It is involved in numerous illustrative and practical issues. The main goal of this paper is to investigate weighted fractional integral inequalities using the Caputo–Fabrizio fractional integral operator with [...] Read more.
The Caputo–Fabrizio fractional integral operator is one of the important notions of fractional calculus. It is involved in numerous illustrative and practical issues. The main goal of this paper is to investigate weighted fractional integral inequalities using the Caputo–Fabrizio fractional integral operator with non-singular e1δδ(ϰs), 0<δ<1. Furthermore, based on a family of n positive functions defined on [0,), we investigate some new extensions of weighted fractional integral inequalities. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
17 pages, 842 KiB  
Article
New Riemann–Liouville Fractional-Order Inclusions for Convex Functions via Interval-Valued Settings Associated with Pseudo-Order Relations
by Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh and Yasser S. Hamed
Fractal Fract. 2022, 6(4), 212; https://doi.org/10.3390/fractalfract6040212 - 9 Apr 2022
Cited by 15 | Viewed by 1684
Abstract
In this study, we focus on the newly introduced concept of LR-convex interval-valued functions to establish new variants of the Hermite–Hadamard (H-H) type and Pachpatte type inequalities for Riemann–Liouville fractional integrals. By presenting some numerical examples, we also verify the correctness of the [...] Read more.
In this study, we focus on the newly introduced concept of LR-convex interval-valued functions to establish new variants of the Hermite–Hadamard (H-H) type and Pachpatte type inequalities for Riemann–Liouville fractional integrals. By presenting some numerical examples, we also verify the correctness of the results that we have derived in this paper. Because the results, which are related to the differintegral of the ϱ1+ϱ22 type, are novel in the context of the LR-convex interval-valued functions, we believe that this will be a useful contribution for motivating future research in this area. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
27 pages, 508 KiB  
Article
Certain New Chebyshev and Grüss-Type Inequalities for Unified Fractional Integral Operators via an Extended Generalized Mittag-Leffler Function
by Wengui Yang
Fractal Fract. 2022, 6(4), 182; https://doi.org/10.3390/fractalfract6040182 - 24 Mar 2022
Cited by 3 | Viewed by 1765
Abstract
In this paper, by adopting the classical method of proofs, we establish certain new Chebyshev and Grüss-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function. The main results are more general and include a large number of available classical [...] Read more.
In this paper, by adopting the classical method of proofs, we establish certain new Chebyshev and Grüss-type inequalities for unified fractional integral operators via an extended generalized Mittag-Leffler function. The main results are more general and include a large number of available classical fractional integral inequalities in the literature. Furthermore, some new fractional integral inequalities similar to the main results can be also obtained by employing the newly introduced generalized fractional integral operators involving the Mittag-Leffler-like function and weighted function. Consequently, their relevance with known inequalities for different kinds of fractional integral operators are pointed out. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
15 pages, 339 KiB  
Article
Riemann–Liouville Fractional Newton’s Type Inequalities for Differentiable Convex Functions
by Thanin Sitthiwirattham, Kamsing Nonlaopon, Muhammad Aamir Ali and Hüseyin Budak
Fractal Fract. 2022, 6(3), 175; https://doi.org/10.3390/fractalfract6030175 - 21 Mar 2022
Cited by 25 | Viewed by 2264
Abstract
In this paper, we prove some new Newton’s type inequalities for differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation. It is also shown that the newly established [...] Read more.
In this paper, we prove some new Newton’s type inequalities for differentiable convex functions through the well-known Riemann–Liouville fractional integrals. Moreover, we prove some inequalities of Riemann–Liouville fractional Newton’s type for functions of bounded variation. It is also shown that the newly established inequalities are the extension of comparable inequalities inside the literature. Finally, we give some examples with graphs and show the validity of newly established inequalities. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
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17 pages, 844 KiB  
Article
New Fractional Integral Inequalities for Convex Functions Pertaining to Caputo–Fabrizio Operator
by Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Muhammad Tariq and Y. S. Hamed
Fractal Fract. 2022, 6(3), 171; https://doi.org/10.3390/fractalfract6030171 - 19 Mar 2022
Cited by 32 | Viewed by 2313
Abstract
In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into [...] Read more.
In this article, a generalized midpoint-type Hermite–Hadamard inequality and Pachpatte-type inequality via a new fractional integral operator associated with the Caputo–Fabrizio derivative are presented. Furthermore, a new fractional identity for differentiable convex functions of first order is proved. Then, taking this identity into account as an auxiliary result and with the assistance of Hölder, power-mean, Young, and Jensen inequality, some new estimations of the Hermite-Hadamard (H-H) type inequality as refinements are presented. Applications to special means and trapezoidal quadrature formula are presented to verify the accuracy of the results. Finally, a brief conclusion and future scopes are discussed. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
20 pages, 351 KiB  
Article
Reverse Minkowski Inequalities Pertaining to New Weighted Generalized Fractional Integral Operators
by Rozana Liko, Pshtiwan Othman Mohammed, Artion Kashuri and Y. S. Hamed
Fractal Fract. 2022, 6(3), 131; https://doi.org/10.3390/fractalfract6030131 - 24 Feb 2022
Cited by 6 | Viewed by 2006
Abstract
In this paper, we obtain reverse Minkowski inequalities pertaining to new weighted generalized fractional integral operators. Moreover, we derive several important special cases for suitable choices of functions. In order to demonstrate the efficiency of our main results, we offer many concrete examples [...] Read more.
In this paper, we obtain reverse Minkowski inequalities pertaining to new weighted generalized fractional integral operators. Moreover, we derive several important special cases for suitable choices of functions. In order to demonstrate the efficiency of our main results, we offer many concrete examples as applications. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
21 pages, 354 KiB  
Article
Some Generalizations of Different Types of Quantum Integral Inequalities for Differentiable Convex Functions with Applications
by Dafang Zhao, Muhammad Aamir Ali, Waewta Luangboon, Hüseyin Budak and Kamsing Nonlaopon
Fractal Fract. 2022, 6(3), 129; https://doi.org/10.3390/fractalfract6030129 - 23 Feb 2022
Cited by 10 | Viewed by 1916
Abstract
In this paper, we prove a new quantum integral equality involving a parameter, left and right quantum derivatives. Then, we use the newly established equality and prove some new estimates of quantum Ostrowski, quantum midpoint, quantum trapezoidal and quantum Simpson’s type inequalities for [...] Read more.
In this paper, we prove a new quantum integral equality involving a parameter, left and right quantum derivatives. Then, we use the newly established equality and prove some new estimates of quantum Ostrowski, quantum midpoint, quantum trapezoidal and quantum Simpson’s type inequalities for q-differentiable convex functions. It is also shown that the newly established inequalities are the refinements of the existing inequalities inside the literature. Finally, some examples and applications are given to illustrate the investigated results. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
20 pages, 326 KiB  
Article
Some Generalized Fractional Integral Inequalities for Convex Functions with Applications
by Dafang Zhao, Muhammad Aamir Ali, Chanon Promsakon and Thanin Sitthiwirattham
Fractal Fract. 2022, 6(2), 94; https://doi.org/10.3390/fractalfract6020094 - 9 Feb 2022
Cited by 5 | Viewed by 1376
Abstract
In this paper, we establish a generalized fractional integrals identity involving some parameters and differentiable functions. Then, we use the newly established identity and prove different generalized fractional integrals inequalities like midpoint inequalities, trapezoidal inequalities and Simpson’s inequalities for differentiable convex functions. Finally, [...] Read more.
In this paper, we establish a generalized fractional integrals identity involving some parameters and differentiable functions. Then, we use the newly established identity and prove different generalized fractional integrals inequalities like midpoint inequalities, trapezoidal inequalities and Simpson’s inequalities for differentiable convex functions. Finally, we give some applications of newly established inequalities in the context of quadrature formulas. Full article
(This article belongs to the Special Issue Fractional Integral Inequalities and Applications)
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