A Quantum Calculus View of Hermite–Hadamard–Jensen–Mercer Inequalities with Applications
Abstract
:1. Introduction and Preliminaries
2. Main Results
3. Applications
- The arithmetic mean:
- The generalized -mean:
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Dragomir, S.S.; Pearce, C. Selected Topics on Hermite-Hadamard Inequality and Applications; Victoria University: Melbourne, Australia, 2000. [Google Scholar]
- Guessab, A. Direct and converse results for generalized multivariate Jensen-type inequalities. J. Nonlinear Convex Anal. 2012, 13, 777–797. [Google Scholar]
- Mercer, A.M. A variant of Jensen’s inequality. J. Inequal. Pure Appl. Math. 2003, 4, 1–6. [Google Scholar]
- Pavić, Z. The Jensen-Mercer inequality with infinite convex combinations. Math. Sci. Appl. E-Notes 2019, 7, 19–27. [Google Scholar]
- Anderson, T.W. Some inequalities for symmetric convex sets with applications. Ann. Stat. 1996, 24, 753–762. [Google Scholar] [CrossRef]
- Boltyanski, V.G.; Castro, J.J. Centrally symmetric convex sets. J. Convex Anal. 2007, 14, 345–351. [Google Scholar]
- Tariboon, J.; Ntouyas, S.K. Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ. 2013, 282, 1–19. [Google Scholar] [CrossRef] [Green Version]
- Alp, N.; Sarıkaya, M.Z.; Kunt, M.; İşcan, İ. q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions. J. King Saud Univ. Sci. 2018, 30, 193–203. [Google Scholar] [CrossRef] [Green Version]
- Noor, M.A.; Noor, K.I.; Awan, M.U. Some Quantum estimates for Hermite-Hadamard inequalities. Appl. Math. Comput. 2015, 251, 675–679. [Google Scholar] [CrossRef]
- Sudsutad, W.; Ntouyas, S.K.; Tariboon, J. Quantum integral inequalities for convex functions. J. Math. Inequal. 2015, 9, 781–793. [Google Scholar] [CrossRef] [Green Version]
- Zhang, Y.; Du, T.-S.; Wang, H.; Shen, Y.-J. Different types of quantum integral inequalities via (α,m)-convexity. J. Inequal. Appl. 2018, 264, 1–24. [Google Scholar] [CrossRef]
- Bermudo, S.; Korus, P.; Valdes, J.N. On q-Hermite-Hadamard inequalities for general convex functions. Acta Math. Hung. 2020, 162, 364–374. [Google Scholar] [CrossRef]
- Budak, H. Some trapezoid and midpoint type inequalities for newly defined quantum integrals. Proyecciones 2021, 40, 199–215. [Google Scholar] [CrossRef]
- Zhao, D.; Gulshan, G.; Ali, M.A.; Nonlaopon, K. Some new midpoint and trapezoidal-type inequalities for general convex functions in q-calculus. Mathematics 2022, 10, 444. [Google Scholar] [CrossRef]
- Du, T.S.; Luo, C.Y.; Yu, B. Certain quantum estimates on the parameterized integral inequalities and their applications. J. Math. Inequal. 2021, 15, 201–228. [Google Scholar] [CrossRef]
- Zhao, D.; Ali, M.A.; Luangboon, W.; Budak, H.; Nonlaopon, K. Some generalizations of different types of quantum integral inequalities for differentiable convex functions with applications. Fractal Fract. 2022, 6, 129. [Google Scholar] [CrossRef]
- Asawasamrit, S.; Ali, M.A.; Ntouyas, S.K.; Tariboon, J. Some parameterized quantum midpoint and quantum trapezoid type inequalities for convex functions with applications. Entropy 2021, 23, 996. [Google Scholar] [CrossRef] [PubMed]
- Asawasamrit, S.; Ali, M.A.; Budak, H.; Ntouyas, S.K.; Tariboon, J. Quantum Hermite-Hadamard and quantum Ostrowski type inequalities for s-convex functions in the second sense with applications. AIMS Math. 2021, 6, 13327–13346. [Google Scholar] [CrossRef]
- Kalsoom, H.; Latif, M.A.; Idrees, M.; Arif, M.; Salleh, Z. Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions. AIMS Math. 2021, 6, 13291–13310. [Google Scholar] [CrossRef]
- Wang, P.P.; Zhu, T.; Du, T.S. Some inequalities using s-preinvexity via quantum calculus. J. Interdiscip. Math. 2021, 24, 613–636. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Bin-Mohsin, B.; Saba, M.; Javed, M.Z.; Awan, M.U.; Budak, H.; Nonlaopon, K. A Quantum Calculus View of Hermite–Hadamard–Jensen–Mercer Inequalities with Applications. Symmetry 2022, 14, 1246. https://doi.org/10.3390/sym14061246
Bin-Mohsin B, Saba M, Javed MZ, Awan MU, Budak H, Nonlaopon K. A Quantum Calculus View of Hermite–Hadamard–Jensen–Mercer Inequalities with Applications. Symmetry. 2022; 14(6):1246. https://doi.org/10.3390/sym14061246
Chicago/Turabian StyleBin-Mohsin, Bandar, Mahreen Saba, Muhammad Zakria Javed, Muhammad Uzair Awan, Hüseyin Budak, and Kamsing Nonlaopon. 2022. "A Quantum Calculus View of Hermite–Hadamard–Jensen–Mercer Inequalities with Applications" Symmetry 14, no. 6: 1246. https://doi.org/10.3390/sym14061246
APA StyleBin-Mohsin, B., Saba, M., Javed, M. Z., Awan, M. U., Budak, H., & Nonlaopon, K. (2022). A Quantum Calculus View of Hermite–Hadamard–Jensen–Mercer Inequalities with Applications. Symmetry, 14(6), 1246. https://doi.org/10.3390/sym14061246