Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
3.6
5.4
Journals
Fractal Fract
Special Issues
Mathematical Inequalities in Fractional Calculus and Applications
share announcement

Mathematical Inequalities in Fractional Calculus 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 March 2024) | Viewed by 16478

Special Issue Editors

Dr. Seth Kermausuor

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL, 36101, USA
Interests: mathematical inequalities; fractional calculus; time scales; modeling; analysis
Special Issues, Collections and Topics in MDPI journals
Dr. Eze Nwaeze

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA
Interests: mathematical inequalities, fractional calculus, time scale theory, and growth properties of complex polynomials
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Inequalities of any type play a very important role in various aspects of mathematical analysis, such as approximation theory and the theory of differential equations. The theory of Fractional Calculus, which deals with the study and applications of derivatives and integrals of arbitrary orders, has gained considerable attention due to its numerous applications in the applied sciences.

In recent years, fractional differential equations have been used frequently in modelling of real systems in numerous fields of applied sciences. To study the existence, uniqueness, and stability of solutions to a system of fractional differential equations inequalities involving derivatives and integrals of arbitrary powers, also known as fractional inequalities, have been used. Additionally, fractional inequalities are used to find upper and lower bounds of solutions to a system of fractional differential equations. In addition, fractional inequalities are also used in the fields of probability, numerical quadrature, and many more. Over the years, many authors have established several generalizations of the various classical inequalities to fractional calculus in the literature.

The goal of this Special Issue is to continue to advance the research on mathematical inequalities in fractional calculus by assembling both original research and review articles on the subject and its related topics. Topics that are invited for submission include (but are not limited to):

  • Fractional integral inequalities;
  • Inequalities of generalized functions in fractional calculus;
  • Q-calculus;
  • Inequalities in fractional calculus on time scales;
  • Fractional order derivatives;
  • Applications of fractional inequalities;
  • Fractals;
  • Non-local mathematical models;
  • Fractional complicated systems

Dr. Seth Kermausuor
Dr. Eze Nwaeze
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 calculus
  • Caputo derivative
  • Riemann-Liouville derivative
  • discrete fractional calculus
  • fractals
  • fractional differential equations
  • mathematical inequalities
  • time scales

Published Papers (15 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

24 pages, 1954 KiB  
Article
New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation
by Tareq Saeed, Eze R. Nwaeze, Muhammad Bilal Khan and Khalil Hadi Hakami
Fractal Fract. 2024, 8(3), 125; https://doi.org/10.3390/fractalfract8030125 - 20 Feb 2024
Viewed by 1101
Abstract
In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ-convexity (LR-ℏ-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville [...] Read more.
In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right ℏ-convexity (LR-ℏ-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated ℏ-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
21 pages, 674 KiB  
Article
Properties and Applications of Symmetric Quantum Calculus
by Miguel Vivas-Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Silvestru Sever Dragomir and Ahmed M. Zidan
Fractal Fract. 2024, 8(2), 107; https://doi.org/10.3390/fractalfract8020107 - 12 Feb 2024
Viewed by 1552
Abstract
Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and [...] Read more.
Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
►▼ Show Figures

Figure 1

18 pages, 321 KiB  
Article
Maclaurin-Type Integral Inequalities for GA-Convex Functions Involving Confluent Hypergeometric Function via Hadamard Fractional Integrals
by Tarek Chiheb, Badreddine Meftah, Abdelkader Moumen and Mohamed Bouye
Fractal Fract. 2023, 7(12), 860; https://doi.org/10.3390/fractalfract7120860 - 02 Dec 2023
Viewed by 997
Abstract
In this manuscript, by using a new identity, we establish some new Maclaurin-type inequalities for functions whose modulus of the first derivatives are GA-convex functions via Hadamard fractional integrals. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
27 pages, 1520 KiB  
Article
Some New Fractional Inequalities for Coordinated Convexity over Convex Set Pertaining to Fuzzy-Number-Valued Settings Governed by Fractional Integrals
by Tareq Saeed, Adriana Cătaș, Muhammad Bilal Khan and Ahmed Mohammed Alshehri
Fractal Fract. 2023, 7(12), 856; https://doi.org/10.3390/fractalfract7120856 - 30 Nov 2023
Cited by 2 | Viewed by 807
Abstract
In this study, we first propose some new concepts of coordinated up and down convex mappings with fuzzy-number values. Then, Hermite–Hadamard-type inequalities via coordinated up and down convex fuzzy-number-valued mapping (coordinated UD-convex FNVMs) are introduced. By [...] Read more.
In this study, we first propose some new concepts of coordinated up and down convex mappings with fuzzy-number values. Then, Hermite–Hadamard-type inequalities via coordinated up and down convex fuzzy-number-valued mapping (coordinated UD-convex FNVMs) are introduced. By taking the products of two coordinated UD-convex FNVMs, Pachpatte-type inequalities are also obtained. Some new conclusions are also derived by making particular decisions with the newly defined inequalities, and it is demonstrated that the recently discovered inequalities are expansions of comparable findings in the literature. It is important to note that the main outcomes are validated using nontrivial examples. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
24 pages, 2145 KiB  
Article
Analysis and Applications of Some New Fractional Integral Inequalities
by Sofia Ramzan, Muhammad Uzair Awan, Silvestru Sever Dragomir, Bandar Bin-Mohsin and Muhammad Aslam Noor
Fractal Fract. 2023, 7(11), 797; https://doi.org/10.3390/fractalfract7110797 - 31 Oct 2023
Viewed by 1094
Abstract
This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications [...] Read more.
This paper presents a novel parameterized fractional integral identity. By using this auxiliary result and the s-convexity property of the mapping, a series of fractional variants of certain classical inequalities, including Simpson’s, midpoint, and trapezoidal-type inequalities, have been derived. Additionally, some applications of our main outcomes to special means of real numbers have been explored. Moreover, we have derived a new generic numerical scheme for solving non-linear equations, demonstrating an application of our main results in numerical analysis. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
►▼ Show Figures

Figure 1

13 pages, 307 KiB  
Article
New Fractional Integral Inequalities via k-Atangana–Baleanu Fractional Integral Operators
by Seth Kermausuor and Eze R. Nwaeze
Fractal Fract. 2023, 7(10), 740; https://doi.org/10.3390/fractalfract7100740 - 08 Oct 2023
Viewed by 840
Abstract
We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some [...] Read more.
We propose the definitions of some fractional integral operators called k-Atangana–Baleanu fractional integral operators. These newly proposed operators are generalizations of the well-known Atangana–Baleanu fractional integral operators. As an application, we establish a generalization of the Hermite–Hadamard inequality. Additionally, we establish some new identities involving these new integral operators and obtained new fractional integral inequalities of the midpoint and trapezoidal type for functions whose derivatives are bounded or convex. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
19 pages, 349 KiB  
Article
On Inequalities and Filtration Associated with the Nonlinear Fractional Operator
by Maryam Nazir, Syed Zakir Hussain Bukhari, Jong-Suk Ro, Fairouz Tchier and Sarfraz Nawaz Malik
Fractal Fract. 2023, 7(10), 726; https://doi.org/10.3390/fractalfract7100726 - 30 Sep 2023
Viewed by 649
Abstract
In this paper, we study a new filtration class MFα,βμ, associated with the filtration of infinitesimal generators, by using the nonlinear fractional differential operator and study certain properties, like sharp Fekete–Szegö inequalities and filtration problems. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
22 pages, 446 KiB  
Article
Advances in Ostrowski-Mercer Like Inequalities within Fractal Space
by Miguel Vivas-Cortez, Muhammad Uzair Awan, Usama Asif, Muhammad Zakria Javed and Hüseyin Budak
Fractal Fract. 2023, 7(9), 689; https://doi.org/10.3390/fractalfract7090689 - 16 Sep 2023
Viewed by 714
Abstract
The main idea of the current investigation is to explore some new aspects of Ostrowski’s type integral inequalities implementing the generalized Jensen–Mercer inequality established for generalized s-convexity in fractal space. To proceed further with this task, we construct a new generalized integral [...] Read more.
The main idea of the current investigation is to explore some new aspects of Ostrowski’s type integral inequalities implementing the generalized Jensen–Mercer inequality established for generalized s-convexity in fractal space. To proceed further with this task, we construct a new generalized integral equality for first-order local differentiable functions, which will serve as an auxiliary result to restore some new bounds for Ostrowski inequality. We establish our desired results by employing the equality, some renowned generalized integral inequalities like Hölder’s, power mean, Yang-Hölder’s, bounded characteristics of the functions and considering generalized s-convexity characteristics of functions. Also, in support of our main findings, we deliver specific applications to means, and numerical integration and graphical visualization are also presented here. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
►▼ Show Figures

Figure 1

14 pages, 343 KiB  
Article
On Bounds of k-Fractional Integral Operators with Mittag-Leffler Kernels for Several Types of Exponentially Convexities
by Ghulam Farid, Hala Safdar Khan, Ferdous M. O. Tawfiq, Jong-Suk Ro and Saira Zainab
Fractal Fract. 2023, 7(8), 617; https://doi.org/10.3390/fractalfract7080617 - 11 Aug 2023
Viewed by 788
Abstract
This paper aims to study the bounds of k-integral operators with the Mittag-Leffler kernel in a unified form. To achieve these bounds, the definition of exponentially (α,h−m)−p-convexity is utilized frequently. In addition, a [...] Read more.
This paper aims to study the bounds of k-integral operators with the Mittag-Leffler kernel in a unified form. To achieve these bounds, the definition of exponentially (α,h−m)−p-convexity is utilized frequently. In addition, a fractional Hadamard type inequality which shows the upper and lower bounds of k-integral operators simultaneously is presented. The results are directly linked with the results of many published articles. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
27 pages, 474 KiB  
Article
Certain New Reverse Hölder- and Minkowski-Type Inequalities for Modified Unified Generalized Fractional Integral Operators with Extended Unified Mittag–Leffler Functions
by Wengui Yang
Fractal Fract. 2023, 7(8), 613; https://doi.org/10.3390/fractalfract7080613 - 09 Aug 2023
Viewed by 717
Abstract
In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities [...] Read more.
In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities in the literature. As applications, the reverse versions of weighted Radon-, Jensen- and power mean-type inequalities for modified unified generalized FIOs with extended unified MLFs are also investigated. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
19 pages, 364 KiB  
Article
Certain Novel Fractional Integral Inequalities via Fuzzy Interval Valued Functions
by Miguel Vivas-Cortez, Rana Safdar Ali, Humira Saif, Mdi Begum Jeelani, Gauhar Rahman and Yasser Elmasry
Fractal Fract. 2023, 7(8), 580; https://doi.org/10.3390/fractalfract7080580 - 28 Jul 2023
Viewed by 900
Abstract
Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. In this article, we elaborate the convexities and pre-invexities in aspects of FIVFs [...] Read more.
Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. In this article, we elaborate the convexities and pre-invexities in aspects of FIVFs and investigate the existence of fuzzy fractional integral operators (FFIOs) having a generalized Bessel–Maitland function as their kernel. Using the class of convexities and pre-invexities FIVFs, we prove some Hermite–Hadamard (H-H) and trapezoid-type inequalities by the implementation of FFIOs. Additionally, we obtain other well known inequalities having significant behavior in the field of fuzzy interval analysis. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
26 pages, 6452 KiB  
Article
Exploration of Hermite–Hadamard-Type Integral Inequalities for Twice Differentiable h-Convex Functions
by Miguel Vivas-Cortez, Muhammad Samraiz, Muhammad Tanveer Ghaffar, Saima Naheed, Gauhar Rahman and Yasser Elmasry
Fractal Fract. 2023, 7(7), 532; https://doi.org/10.3390/fractalfract7070532 - 07 Jul 2023
Cited by 3 | Viewed by 1814
Abstract
The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set [...] Read more.
The significance of fractional calculus cannot be underestimated, as it plays a crucial role in the theory of inequalities. In this paper, we study a new class of mean-type inequalities by incorporating Riemann-type fractional integrals. By doing so, we discover a novel set of such inequalities and analyze them using different mathematical identities. This particular class of inequalities is introduced by employing a generalized convexity concept. To validate our work, we create visual graphs and a table of values using specific functions to represent the inequalities. This approach allows us to demonstrate the validity of our findings and further solidify our conclusions. Moreover, we find that some previously published results emerge as special consequences of our main findings. This research serves as a catalyst for future investigations, encouraging researchers to explore more comprehensive outcomes by using generalized fractional operators and expanding the concept of convexity. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
►▼ Show Figures

Figure 1

18 pages, 344 KiB  
Article
On Further Inequalities for Convex Functions via Generalized Weighted-Type Fractional Operators
by Çetin Yıldız, Gauhar Rahman and Luminiţa-Ioana Cotîrlă
Fractal Fract. 2023, 7(7), 513; https://doi.org/10.3390/fractalfract7070513 - 28 Jun 2023
Viewed by 594
Abstract
Several inequalities for convex functions are derived in this paper using the monotonicity properties of functions and a generalized weighted-type fractional integral operator, which allows the integration of a function κ with respect to another function in fractional order. Additionally, it is clear [...] Read more.
Several inequalities for convex functions are derived in this paper using the monotonicity properties of functions and a generalized weighted-type fractional integral operator, which allows the integration of a function κ with respect to another function in fractional order. Additionally, it is clear that the results were generalizations of the previously presented findings. In addition, different types of inequalities are obtained using the basic features of mathematical analysis. Finally, we believe that the methodology used in this work will inspire additional research in this field. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
16 pages, 335 KiB  
Article
Several Quantum Hermite–Hadamard-Type Integral Inequalities for Convex Functions
by Loredana Ciurdariu and Eugenia Grecu
Fractal Fract. 2023, 7(6), 463; https://doi.org/10.3390/fractalfract7060463 - 07 Jun 2023
Cited by 5 | Viewed by 1006
Abstract
The aim of this study was to present several improved quantum Hermite–Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some [...] Read more.
The aim of this study was to present several improved quantum Hermite–Hadamard-type integral inequalities for convex functions using a parameter. Thus, a new quantum identity is proven to be used as the main tool in the proof of our results. Consequently, in some special cases several new quantum estimations for q-midpoints and q-trapezoidal-type inequalities are derived with an example. The results obtained could be applied in the optimization of several economic geology problems. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
►▼ Show Figures

Figure 1

19 pages, 361 KiB  
Article
Some Novel Inequalities for LR-(k,h-m)-p Convex Interval Valued Functions by Means of Pseudo Order Relation
by Vuk Stojiljković, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby and Stojan Radenović
Fractal Fract. 2022, 6(12), 726; https://doi.org/10.3390/fractalfract6120726 - 09 Dec 2022
Cited by 10 | Viewed by 1344
Abstract
In this paper, a new type of convexity is defined, namely, the left–right-(k,h-m)-p IVM (set-valued function) convexity. Utilizing the definition of this new convexity, we prove the Hadamard inequalities for noninteger Katugampola integrals. These inequalities generalize the noninteger Hadamard inequalities for a convex [...] Read more.
In this paper, a new type of convexity is defined, namely, the left–right-(k,h-m)-p IVM (set-valued function) convexity. Utilizing the definition of this new convexity, we prove the Hadamard inequalities for noninteger Katugampola integrals. These inequalities generalize the noninteger Hadamard inequalities for a convex IVM, (p,h)-convex IVM, p-convex IVM, h-convex, s-convex in the second sense and many other related well-known classes of functions implicitly. An apt number of numerical examples are provided as supplements to the derived results. Full article
(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-15
Back to TopTop