A note on the generalization of Minkowski's and some other types of integral inequalities via modified AB-fractional operator

Volume 40, Issue 1, pp 38--48 https://dx.doi.org/10.22436/jmcs.040.01.03

Publication Date: May 30, 2025 Submission Date: September 04, 2024 Revision Date: November 02, 2024 Accteptance Date: March 27, 2025

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 117 Downloads
• 455 Views

Authors

G. Rahman - Department of Mathematics and Statistics, Hazara University, Bosan Road, Mansehra 21300, Pakistan. C. Yildiz - Department of Mathematics, K.K. Education Faculty, Atatürk University, Bosan Road, 25240, Erzurum, Turkey. M. Samraiz - Department of Mathematics, University of Sargodha, Sargodha, Pakistan. S. S. Aiadi - Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia. N. Mlaiki - Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia.

Abstract

Many researchers are attracted to the AB-fractional integral operators for the investigation of a wide range of problems in divers fields. In this paper, we take into consideration the definition of modified AB-fractional integral operators recently proposed by [W. H. Huang, M. Samraiz, A. Mehmood, D. Baleanu, G. Rahman, S. Naheed, Alex. Eng. J., \(\bf 75\) (2023), 639--648]. With the modified AB-fractional integral operators, we propose a generalization of the reverse Minkowski inequality. We also use the modified AB-fractional integral operator to develop some more related integral inequalities. The inequalities in this paper are more general as compared to the existing inequalities. These inequalities extend fractional versions of Minkowski-type inequalities that are both previously known and novel.

Share and Cite

Keywords

• Minkowski inequality
• AB-fractional operator
• modified AB-fractional integral operator

MSC

•  26D15
•  26A33
•  35J05

References

• [1] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 11 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Abdeljawad, Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals, Chaos, 29 (2019), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag- Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098–1107
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] T. Abdeljawad, D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Differ. Equ., 2018 (2018), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] P. Agarwal, M. Jleli, M. Tomar, Certain Hermite-Hadamard type inequalities via generalized k-fractional integrals, J. Inequal. Appl., 2017 (2017), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] W. F. S. Ahmed, D. D. Pawar, W. D., Patil, Solution of fractional kinetic equations involving Laguerre polynomials via Sumudu transform, J. Funct. Spaces, 2024 (2024), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [7] M. Aldhaifallah, M. Tomar, K. S. Nisar, S. D. Purohit, Some new inequalities for (k, s)-fractional integrals, J. Nonlinear Sci. Appl., 9 (2016), 5374–5381
  • View Article
  • MathSciNet
  • Google Scholar


• [8] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769
  • View Article
  • Google Scholar


• [9] D. Baleanu, R. P. Agarwal, H. Khan, R. A. Khan, H. Jafari, On the existence of solution for fractional differential equations of order 3 < δ1 ⩽ 4, Adv. Differ. Equ., 2015 (2015), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [10] D. Baleanu, R. P. Agarwal, H. Mohammadi, S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl., 2013 (2013), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] D. Baleanu, A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 444–462
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] D. Baleanu, H. Khan, H. Jafari, R. A. Khan, M. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Differ. Equ., 2015 (2015), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] D. B˘aleanu, O. G. Mustafa, P. R. Agarwal, An existence result for a superlinear fractional differential equation, Appl. Math. Lett., 23 (2010), 1129–1132
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. B˘aleanu, O. G. Mustafa, R. P. Agarwal, On the solution set for a class of sequential fractional differential equations, J. Phys. A, 43 (2010), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] L. Bougoffa, On Minkowski and Hardy integral inequalities, JIPAM. J. Inequal. Pure Appl. Math., 7 (2006), 3 pages
  • View Article
  • Google Scholar
  • MATH


• [16] M. Ça˘ glar, K. R. Karthikeyan, G. Murugusundaramoorthy, Inequalities on a class of analytic functions defined by generalized Mittag-Leffler function, Filomat, 37 (2023), 6277–6288
  • View Article
  • Google Scholar


• [17] G. Farid, K. A. Khan, N. Latif, A. U. Rehman, S. Mehmood, General fractional integral inequalities for convex and m-convex functions via an extended generalized Mittag-Leffler function, J. Inequal. Appl., 2018 (2018), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] W. H. Huang, M. Samraiz, A. Mehmood, D. Baleanu, G. Rahman, S. Naheed, Modified Atangana-Baleanu fractional operators involving generalized Mittag-Leffler function, Alex. Eng. J., 75 (2023), 639–648
  • View Article
  • Google Scholar


• [19] H. Khan, T. Abdeljawad, C. Tunç, A. Alkhazzan, A. Khan, Minkowski’s inequality for the AB-fractional integral operator, J. Inequal. Appl., 2019 (2019), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] H. Khan, C. Tunç, A. Alkhazan, B. Ameen, A. Khan, A generalization of Minkowski’s inequality by Hahn integral operator, J. Taibah Univ. Sci., 12 (2018), 506–513
  • View Article
  • Google Scholar


• [21] K. Mehrez, P. Agarwal, New Hermite-Hadamard type integral inequalities for convex functions and their applications, J. Comput. Appl. Math., 350 (2019), 274–285
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] P. A. Naik, M. Yavuz, S. Qureshi, M. D. Naik, K. M. Owolabi, A. Soomro, A. H. Ganie, Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment, Comput. Methods Programs Biomed., 254 (2024),
  • View Article
  • Google Scholar


• [23] A. Nazir, G. Rahman, A. Ali, S. Naheed, K. S. Nisar, W. Albalawi, H. Y. Zahran, On generalized fractional integral with multivariate Mittag-Leffler function and its applications, Alex. Eng. J., 61 (2022), 9187–9201
  • View Article
  • Google Scholar


• [24] M. J. Park, O. M. Kwon, J. H. Ryu, Generalized integral inequality: application to time-delay systems, Appl. Math. Lett., 77 (2018), 6–12
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] G. Rahman, T. Abdeljawad, F. Jarad, K. S. Nisar, Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and their Applications, Mathematics, 8 (2020), 19 pages
  • View Article
  • Google Scholar


• [26] G. Rahman, K. S. Nisar, A. Ghaffar, F. Qi, Some inequalities of the Grüss type for conformable k-fractional integral operators, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., 114 (2020),
  • View Article
  • Google Scholar
  • MATH


• [27] G. Rahman, K. S. Nisar, F. Qi, Some new inequalities of the Grüss type for conformable fractional integrals, AIMS Math., 3 (2018), 575–583
  • View Article
  • Google Scholar
  • MATH


• [28] G. Rahman, Z. Ullah, A. Khan, E. Set, K. S. Nisar, Certain Chebyshev type inequalities involving fractional conformable integral operators, Mathematics, 7 (2019), 9 pages
  • View Article
  • Google Scholar


• [29] M. Samraiz, A. Mehmood, S. Naheed, G. Rahman, A. Kashuri, K. Nonlaopon, On Novel Fractional Operators Involving the Multivariate Mittag-Leffler Function, Mathematics, 10 (2022), 19 pages
  • View Article
  • Google Scholar


• [30] M. Samraiz, Z. Perveen, T. Abdeljawad, S. Iqbal, S. Naheed, On Certain Fractional Calculus Operators and Their Applications in Mathematical Physics, Phys. Scr., 95 (2020),
  • View Article
  • Google Scholar


• [31] M. Samraiz, Z. Perveen, G. Rahman, K. S. Nisar, D. Kumar, On (k, s)-Hilfer Prabhakar Fractional Derivative with Applications in Mathematical Physics, Front. Phys., 8 (2020), 1–9
  • View Article
  • Google Scholar


• [32] M. Samraiz, M. Umer, T. Abdeljawad, S. Naheed, G. Rahman, K. Shah, On Riemann-type weighted fractional operators and solutions to Cauchy problems, Comput. Model. Eng. Sci., 136 (2023), 901–919
  • View Article
  • Google Scholar


• [33] M. Z. Sarikaya, T. Tunc, H. Budak, On generalized some integral inequalities for local fractional integrals, Appl. Math. Comput., 276 (2016), 316–323
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] E. Set, M. A. Noor, M. U. Awan, A. Gözpinar, Generalized Hermite-Hadamard type inequalities involving fractional integral operators, J. Inequal. Appl., 2017 (2017), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [35] Y. Shuang, F. Qi, Integral inequalities of Hermite-Hadamard type for extended s-convex functions and applications, Mathematics, 6 (2018), 12 pages
  • View Article
  • Google Scholar


• [36] A. Tassaddiq, R. Srivastava, R. Alharbi, R. M. Kasmani, S. Qureshi, An application of multiple Erdélyi–Kober fractional integral operators to establish new inequalities involving a general class of functions, Fractal Fract., 8 (2024), 15 pages
  • View Article
  • Google Scholar