Integral inequalities of Hermite-Hadamard type via $ q-h $ integrals

Dong Chen; Matloob Anwar; Ghulam Farid; Waseela Bibi; Dong Chen; Matloob Anwar; Ghulam Farid; Waseela Bibi

doi:10.3934/math.2023826

AIMS Mathematics

2023, Volume 8, Issue 7: 16165-16174. doi: 10.3934/math.2023826

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Research article

Integral inequalities of Hermite-Hadamard type via $ q-h $ integrals

1.
School of Electronic Information and Electrical Engineering, Chengdu University, Chengdu, China
2.
Key Laboratory of Pattern Recognition and Intelligent Information Processing of Sichuan, Chengdu University, Chengdu, China
3.
School of Natural Sciences, NUST, Islamabad, Pakistan
4.
Department of Mathematics, COMSATS Islamabad, Attock Campus, Pakistan

Received: 29 January 2023 Revised: 26 March 2023 Accepted: 09 April 2023 Published: 06 May 2023
MSC : 26A33, 26A51, 33E12

The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for $ q-h $-integrals using properties of convex functions. Inequalities for $ q $-integrals that have been published in recent years can be extracted from the main results of this paper.
- Hermite-Hadamard inequality,
- convex function,
- $ q-h $-integral,
- $ q-h $-derivative,
- $ q $-integral
Citation: Dong Chen, Matloob Anwar, Ghulam Farid, Waseela Bibi. Integral inequalities of Hermite-Hadamard type via $ q-h $ integrals[J]. AIMS Mathematics, 2023, 8(7): 16165-16174. doi: 10.3934/math.2023826

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Abstract

The well-known Hermite-Hadamard inequality for convex functions is extensively studied for different kinds of integrals and derivatives. This paper investigates some of its variants for $ q-h $-integrals using properties of convex functions. Inequalities for $ q $-integrals that have been published in recent years can be extracted from the main results of this paper.

References

[1]	M. Z. Sarikaya, E. Set, H. Yaldiz, N. Başak, Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model., 57 (2013), 2403–2407. https://doi.org/10.1016/j.mcm.2011.12.048 doi: 10.1016/j.mcm.2011.12.048
[2]	M. Andrić, G. Farid, J. Pečarić, Analytical inequalities for fractional calculus operators and the Mittag-Leffler function, Zagreb: Element, 2021.
[3]	N. Alp, M. Z. Sarikaya, M. Kunt, I. Iscan, $q$-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. https://doi.org/10.1016/j.jksus.2016.09.007 doi: 10.1016/j.jksus.2016.09.007
[4]	S. Bermudo, P. Kórus, J. E. Nápoles Valdés, On $q$-Hermite–Hadamard inequalities for general convex functions, Acta Math. Hungar., 162 (2020), 364–374. https://doi.org/10.1007/s10474-020-01025-6 doi: 10.1007/s10474-020-01025-6
[5]	M. A. Khan, N. Mohammad, E. R. Nwaeze, Y.-M. Chu, Quantum Hermite–Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020), 99. https://doi.org/10.1186/s13662-020-02559-3 doi: 10.1186/s13662-020-02559-3
[6]	G. Farid, M. Anwar, M. Shoaib, On generalizations of $q$- and $h$-integrals and some related inequalities, submitted for publication.
[7]	J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 121. https://doi.org/10.1186/1029-242X-2014-121 doi: 10.1186/1029-242X-2014-121
[8]	Y. Liu, G. Farid, D. Abuzaid, K. Nonlaopon, On $q$-Hermite-Hadamard inequalities via $q-h$-integrals, Symmetry, 14 (2022), 2648. https://doi.org/10.3390/sym14122648 doi: 10.3390/sym14122648
[9]	V. Kac, P. Cheung, Quantum calculus, New York: Springer, 2002. https://doi.org/10.1007/978-1-4613-0071-7
[10]	T. Ernst, A comprehensive treatment of $q$-calculus, Basel: Birkhäuser, 2012. https://doi.org/10.1007/978-3-0348-0431-8
[11]	S. Marinković, P. Rajković, M. Stanković, The inequalities for some types $q$-integrals, Comput. Math. Appl., 56 (2008), 2490–2498. https://doi.org/10.1016/j.camwa.2008.05.035 doi: 10.1016/j.camwa.2008.05.035
[12]	Y. Miao, F. Qi, Several $q$-integral inequalities, J. Math. Inequal., 3 (2009), 115–121. http://dx.doi.org/10.7153/jmi-03-11 doi: 10.7153/jmi-03-11
[13]	K. Brahim, N. Bettaibi, M. Sellami, On some Feng Qi type $q$-integral inequalities, Journal of Inequalities in Pure and Applied Mathematics, 9 (2008), 43.
[14]	H. Gauchman, Integral inequalities in $q$-calculus, Comput. Math. Appl., 47 (2004), 281–300. https://doi.org/10.1016/S0898-1221(04)90025-9 doi: 10.1016/S0898-1221(04)90025-9
[15]	A. A. El-Deeb, J. Awrejcewicz, Ostrowski-Trapezoid-Grüss-type on $(q, \omega)$-Hahn difference operator, Symmetry, 14 (2022), 1776. https://doi.org/10.3390/sym14091776 doi: 10.3390/sym14091776
[16]	S. I. Butt, H. Budak, K. Nonlaopon, New quantum Mercer estimates of Simpson–Newton-like inequalities via convexity, Symmetry, 14 (2022), 1935. https://doi.org/10.3390/sym14091935 doi: 10.3390/sym14091935

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