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Mathematical Inequalities and Fractional Calculus
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Mathematical Inequalities and Fractional Calculus

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 30 June 2025 | Viewed by 3623

Special Issue Editors

Dr. Péter Kórus

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Guest Editor
Institute of Applied Pedagogy, Juhász Gyula Faculty of Education, University of Szeged, H-6725 Szeged, Hungary
Interests: mathematical analysis; convex functions; fractional integrals
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Juan Eduardo Nápoles Valdés

E-Mail Website
Guest Editor
Facultad de Ciencias Exactas y Naturales y Agrimensura, Universidad Nacional del Nordeste, Av. Libertad 5450, Corrientes 3400, Argentina
Interests: fractional calculus; generalized calculus; integral inequalities; qualitative theory of ordinary differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Integral inequalities are a fundamental concept in calculus and mathematics in general. They have great importance in various fields, both theoretical and applied, such as the analysis of functions, measure theory, functional analysis, optimization and control, initial and boundary value problems, and error estimation.

In recent years, interest in the study of classical inequalities has increased. Broadly speaking, integral inequalities can be categorized into the following groups:

  • Classical integral inequalities.

Inequalities that do not involve the notion of convexity: Hölder’s inequality, the power mean inequality, Minkowski’s inequality, Chebyshev’s inequality, Grüss’ inequality, and Wirtinger’s inequality.

Inequalities that use the notion of convexity: Simpson's inequality, Jensen’s (Jensen–Mercer) inequality, and the Hermite–Hadamard and Hermite–Hadamard–Fejér inequalities.

  • Auxiliary integral inequalities: Hölder’s inequality, the power mean inequality, and Minkowski’s inequality.
  • Integral inequalities that involve products of integrals: Chebyshev’s inequality and Grüss’ inequality.
  • Integral inequalities that involve derivatives: Wirtinger’s inequality and Simpson's inequality.

Generalizations of the above inequalities are often applied to integral operators associated with different types of fractional integrals and derivatives, such as the Hadamard, Riemann–Liouville, Weil, Erdelyi–Kober, Katugampola integrals and other types defined by different mathematicians. These results have demonstrated their usefulness and potential in the modeling of different processes and phenomena.

We cordially invite interested researchers to contribute original and high-quality research on the aforementioned topics to this Special Issue.

Dr. Péter Kórus
Prof. Dr. Juan Eduardo Nápoles Valdés
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • inequalities
  • integral inequalities
  • fractional calculus
  • q-calculus
  • fractional integral operator
  • fractional differential operator
  • fractional differential equation
  • fractional integral equation

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Published Papers (4 papers)

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Research

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21 pages, 2357 KiB  
Article
On the Martínez–Kaabar Fractal–Fractional Reduced Pukhov Differential Transformation and Its Applications
by Francisco Martínez and Mohammed K. A. Kaabar
Mathematics 2025, 13(3), 352; https://doi.org/10.3390/math13030352 - 22 Jan 2025
Viewed by 880
Abstract
This paper addresses the extension of the Martinez–Kaabar fractal–fractional calculus (simply expressed as MK calculus) to the context of reduced differential transformation, with applications to the solution of some partial differential equations. Since this differential transformation is derived from the Taylor series expansion [...] Read more.
This paper addresses the extension of the Martinez–Kaabar fractal–fractional calculus (simply expressed as MK calculus) to the context of reduced differential transformation, with applications to the solution of some partial differential equations. Since this differential transformation is derived from the Taylor series expansion of real-valued functions of several variables, it is necessary to develop this theory in the context of such functions. Firstly, classical elements of the analysis of functions of several real variables are introduced, such as the concept of partial derivative and Clairaut’s theorem, in terms of the MK partial α,γ-derivative. Next, we establish the fractal–fractional (FrFr) Taylor formula with Lagrange residue and discuss a sufficient condition for a function of class Cα,γ∞ on an open and bounded set D⊂R2 to be expanded into a convergent infinite series, the so-called FrFr Taylor series. The theoretical study is completed by defining the FrFr reduced differential transformation and establishing its fundamental properties, which will allow the construction of the FrFr reduced Pukhov differential transformation method (FrFrRPDTM). Based on the previous results, this new technique is applied to solve interesting non-integer order linear and non-linear partial differential equations that incorporate the fractal effect. Finally, the results show that the FrFrRPDTM represents a simple instrument that provides a direct, efficient, and effective solution to problems involving this class of partial differential equations. Full article
(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)
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20 pages, 1314 KiB  
Article
Upper Bounds for the Remainder Term in Boole’s Quadrature Rule and Applications to Numerical Analysis
by Muhammad Zakria Javed, Muhammad Uzair Awan, Bandar Bin-Mohsin and Savin Treanţă
Mathematics 2024, 12(18), 2920; https://doi.org/10.3390/math12182920 - 20 Sep 2024
Cited by 1 | Viewed by 877
Abstract
In the current study, we compute some upper bounds for the remainder term of Boole’s quadrature rule involving convex mappings. First, we build a new identity for first-order differentiable mapping, an auxiliary result to establish our required estimates. We provide several upper bounds [...] Read more.
In the current study, we compute some upper bounds for the remainder term of Boole’s quadrature rule involving convex mappings. First, we build a new identity for first-order differentiable mapping, an auxiliary result to establish our required estimates. We provide several upper bounds by utilizing the identity, convexity property, and bounded property of mappings and some well-known inequalities. Moreover, based on our primary findings, we deliver applications to the means, quadrature rule, special mappings, and non-linear analysis by developing a novel iterative scheme with cubic order of convergence. To the best of our knowledge, the current study is the first attempt to derive upper bounds for Boole’s scheme involving convex mappings. Full article
(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)
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23 pages, 384 KiB  
Article
Some Simpson- and Ostrowski-Type Integral Inequalities for Generalized Convex Functions in Multiplicative Calculus with Their Computational Analysis
by Xinlin Zhan, Abdul Mateen, Muhammad Toseef and Muhammad Aamir Ali
Mathematics 2024, 12(11), 1721; https://doi.org/10.3390/math12111721 - 31 May 2024
Cited by 7 | Viewed by 987
Abstract
Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas [...] Read more.
Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus. Full article
(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)
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Review

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12 pages, 267 KiB  
Review
A Review of the Chebyshev Inequality Pertaining to Fractional Integrals
by Péter Kórus and Juan Eduardo Nápoles Valdés
Mathematics 2025, 13(7), 1137; https://doi.org/10.3390/math13071137 - 30 Mar 2025
Viewed by 176
Abstract
In this article, we give a brief review of a well-known integral inequality that gives information about the integral of the product of two functions using synchronous functions, the Chebyshev inequality. We have compiled the most relevant information about fractional and generalized integrals, [...] Read more.
In this article, we give a brief review of a well-known integral inequality that gives information about the integral of the product of two functions using synchronous functions, the Chebyshev inequality. We have compiled the most relevant information about fractional and generalized integrals, which are one of the most dynamic topics in today’s mathematical sciences. After presenting the classical formulation of the inequality using Lebesgue integrable functions, the most general results known from the literature are collected in an attempt to present the reader with a current overview of this research topic. Full article
(This article belongs to the Special Issue Mathematical Inequalities and Fractional Calculus)
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