Some quantum corrected dual Euler-Simpson type inequalities for \(q\)-differentiable convex functions

Volume 40, Issue 4, pp 548--565 https://dx.doi.org/10.22436/jmcs.040.04.08

Publication Date: August 20, 2025 Submission Date: March 21, 2025 Revision Date: April 23, 2025 Accteptance Date: July 12, 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)

• 205 Downloads
• 694 Views

Authors

A. Moumen - Department of Mathematics, Faculty of Sciences, University of Ha'il, Ha'il 55473, Saudi Arabia. R. Debbar - University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria. B. Meftah - Laboratory of Analysis and Control of Differential Equations , University of 8 May 1945 Guelma, P.O. Box 401, 24000 Guelma, Algeria. M. Bouye - Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia.

Abstract

The calculus of integrals is used to solve the majority of physics and engineering issues, which are frequently not immediately solved. This compels us to use approximation techniques, the selection of which is based on the class of functions that meet the necessary criteria as well as known points. Within the context of quantum calculus, we present an assessment of the error of the well-known corrected dual Euler-Simpson quadrature rule in this paper. As an auxiliary result, we create a new quantum identity. Using this identity, we prove several quantum-corrected dual Euler-Simpson type integral inequalities for functions with convex \(q\)-derivatives. We allow q to go towards \(1^{-}\) in order to obtain the classical inequalities. We provide a few applications to wrap up this research.

Share and Cite

Keywords

• Corrected dual Euler-Simpson inequality
• convex functions
• \(q\)-derivative
• \(q\)-integration

MSC

•  81P68
•  26D10
•  26A51

References

• [1] A. Agli´c Aljinovi´c, D. Kovaˇcevi´c, M. Puljiz, A. Žgalji´c Keko, Sharp trapezoid inequality for quantum integral operator, Filomat, 36 (2022), 5653–5664
  • View Article
  • MathSciNet
  • Google Scholar


• [2] P. Agarwal, S. S. Dragomir, J. Park, S. Jain, q-integral inequalities associated with some fractional q-integral operators, J. Inequal. Appl., 2015 (2015), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] N. Alp, M. Z. Sarıkaya, M. Kunt, ˙I. ˙I ¸scan, q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, J. King Saud Univ.-Sci., 30 (2018), 193–203
  • View Article
  • Google Scholar


• [4] M. U. Awan, S. Talib, A. Kashuri, M. A. Noor, K. I. Noor, Y.-M. Chu, A new q-integral identity and estimation of its bounds involving generalized exponentially μ-preinvex functions, Adv. Difference Equ., 2020 (2020), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] H. Budak, P. Kösem, H. Kara, On new Milne-type inequalities for fractional integrals, J. Inequal. Appl., 2023 (2023), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] S. I. Butt, H. Budak, K. Nonlaopon, New quantum Mercer estimates of Simpson-Newton like inequalities via convexity, Symmetry, 14 (2022), 21 pages
  • View Article
  • Google Scholar


• [7] S. I. Butt, H. Budak, K. Nonlaopon, New variants of quantum midpoint-type inequalities, Symmetry, 14 (2022), 16 pages
  • View Article
  • Google Scholar


• [8] Y. Deng, M. U. Awan, S. Wu, Quantum integral inequalities of Simpson-type for strongly preinvex functions, Mathematics, 7 (2019), 14 pages
  • View Article
  • Google Scholar


• [9] T. Du, C. Luo, B. Yu, Certain quantum estimates on the parameterized integral inequalities and their applications, J. Math. Inequal., 15 (2021), 201–228
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] S. Erden, S. Iftikhar, M. R. Delavar, P. Kumam, P. Thounthong, W. Kumam, On generalizations of some inequalities for convex functions via quantum integrals, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), 15 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] I. Franji´c, J. Peˇcari´c, On corrected dual Euler-Simpson formulae, Soochow J. Math., 32 (2006), 575–587
  • View Article
  • Google Scholar
  • MATH


• [12] F. H. Jackson, On a q-Definite Integrals, Quart. J. Pure Appl. Math., 41 (1910), 193–203
  • View Article
  • Google Scholar
  • MATH


• [13] S. Jain, P. Agarwal, B. Ahmad, S. K. Q. Al-Omari, Certain recent fractional integral inequalities associated with the hypergeometric operators, J. King Saud Univ.-Sci., 28 (2016), 82–86
  • View Article
  • Google Scholar


• [14] S. Jain, K. Mehrez, D. Baleanu, P. Agarwal, Certain Hermite–Hadamard inequalities for logarithmically convex functions with applications, Mathematics, 7 (2019), 12 pages
  • View Article
  • Google Scholar


• [15] V. Kac, P. Cheung, Quantum calculus, Springer-Verlag, New York (2002)
  • View Article
  • MathSciNet
  • Google Scholar


• [16] A. Ditta, A. Nosheen, K. M. Awan, R. Matendo Mabela, Ostrowski type inequalities for s-convex functions via q-integrals, J. Funct. Spaces, 2022 (2022), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] A. Lakhdari, B. Bin-Mohsin, F. Jarad, H. Xu, B. Meftah, A parametrized approach to generalized fractional integral inequalities: Hermite–Hadamard and Maclaurin variants, J. King Saud Uni.-Sci., 36 (2024), 9 pages
  • View Article
  • Google Scholar


• [18] W. Liu, H. Zhuang, Some quantum estimates of Hermite-Hadamard inequalities for convex functions, J. Appl. Anal. Comput., 7 (2017), 501–522
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] B. Meftah, A. Lakhdari, Dual Simpson type inequalities for multiplicatively convex functions, Filomat, 37 (2023), 7673–7683
  • View Article
  • Google Scholar


• [20] B. Meftah, A. Souahi, M. Merad, Quantum Simpson like type inequalities for q-differentiable convex functions, J. Anal., 32 (2024), 1331–1365
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] M. A. Noor, M. U. Awan, K. I. Noor, Some new q-estimates for certain integral inequalities, Facta Univ. Ser. Math. Inform., 31 (2016), 801–813
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] M. A. Noor, G. Cristescu, M. U. Awan, Bounds having Riemann type quantum integrals via strongly convex functions, Studia Sci. Math. Hungar., 54 (2017), 221–240
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] M. A. Noor, K. I. Noor, M. U. Awan, quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675–679
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] Y. Peng, T. Du, Fractional Maclaurin-type inequalities for multiplicatively convex functions and multiplicatively Pfunctions, Filomat, 37 (2023), 9497–9509
  • View Article
  • Google Scholar


• [25] Saleh, A. Lakhdari, T. Abdeljawad, B. Meftah, On fractional biparameterized Newton-type inequalities, J. Inequal. Appl., 2023 (2023), 18 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] W. Saleh, B. Meftah, A. Lakhdari, Quantum dual Simpson type inequalities for q-differentiable convex functions, Int. J. Nonlinear Anal. Appl., 14 (2023), 63–76
  • View Article
  • Google Scholar


• [27] J. Soontharanon, M. A. Ali, H. Budak, K. Nonlaopon, Z. Abdullah, Simpson’s and Newton’s type Inequalities for (α,m)-convex functions via quantum calculus, Symmetry, 14 (2022), 13 pages
  • View Article
  • Google Scholar


• [28] W. Sudsutad, S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, J. Math. Inequal., 9 (2015), 781–793
  • View Article
  • MathSciNet
  • Google Scholar


• [29] J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ., 2013 (2013), 19 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 13 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] M. Tunç, E. Göv, S. Balgeçti, Simpson type quantum integral inequalities for convex functions, Miskolc Math. Notes, 19 (2018), 649–664
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] P.-P. Wang, T. Zhu, T.-S. Du, Some inequalities using s-preinvexity via quantum calculus, Interdiscip. Math., 24 (2021), 613–636
  • View Article
  • Google Scholar


• [33] Y. Zhang, T.-S. Du, H.Wang, Y.-J. Shen, Different types of quantum integral inequalities via (α,m)-convexity, J. Inequal. Appl., 2018 (2018), 24 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] W. S. Zhu, B. Meftah, H. Xu, F. Jarad, A. Lakhdari, On parameterized inequalities for fractional multiplicative integrals, Demonstr. Math., 57 (2024), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH