Post-quantum trapezoid type inequalities

Muhammad Amer Latif; Mehmet Kunt; Sever Silvestru Dragomir; İmdat İşcan; Muhammad Amer Latif; Mehmet Kunt; Sever Silvestru Dragomir; İmdat İşcan

doi:10.3934/math.2020258

AIMS Mathematics

2020, Volume 5, Issue 4: 4011-4026. doi: 10.3934/math.2020258

Previous Article Next Article

Research article

Post-quantum trapezoid type inequalities

1.
Department of Basic Sciences, Deanship of Preparatory Year, King Faisal University, Hofuf 31982, Al-Hasa, Saudi Arabia
2.
Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080, Trabzon, Turkey
3.
Mathematics, College of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
4.
School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
5.
Department of Mathematics, Faculty of Sciences and Arts, Giresun University, 28200, Giresun, Turkey

Received: 17 October 2019 Accepted: 16 April 2020 Published: 27 April 2020
MSC : 26A51, 26D15, 34A08

In this study, the assumption of being differentiable for the convex function f in the (p, q)-Hermite-Hadamard inequality is removed. A new identity for the right-hand part of (p, q)-Hermite-Hadamard inequality is proved. By using established identity, some (p, q)-trapezoid integral inequalities for convex and quasi-convex functions are obtained. The presented results in this work extend some results from the earlier research.
- Hermite-Hadamard inequality,
- q-integral inequalities,
- (p, q)-integral inequalities,
- (p, q)-derivative,
- (p, q)-integration,
- convexity,
- quasi-convexity
Citation: Muhammad Amer Latif, Mehmet Kunt, Sever Silvestru Dragomir, İmdat İşcan. Post-quantum trapezoid type inequalities[J]. AIMS Mathematics, 2020, 5(4): 4011-4026. doi: 10.3934/math.2020258

Related Papers:

Abstract

In this study, the assumption of being differentiable for the convex function f in the (p, q)-Hermite-Hadamard inequality is removed. A new identity for the right-hand part of (p, q)-Hermite-Hadamard inequality is proved. By using established identity, some (p, q)-trapezoid integral inequalities for convex and quasi-convex functions are obtained. The presented results in this work extend some results from the earlier research.

References

[1]	M. Alomari, M. Darus, S. S. Dragomir, Inequalities of Hermite-Hadamard's type for functions whose derivatives absolute values are quasi-convex, RGMIA Res. Rep. Coll., 12 (2009), 1-11.
[2]	N. Alp, M. Z. Sarıkaya, M. Kunt, et al. 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. doi: 10.1016/j.jksus.2016.09.007
[3]	N. Alp, M. Z. Sarıkaya, A new definition and properties of quantum integral which calls q-integral, Konuralp J. Math., 5 (2017), 146-159.
[4]	A. G. Azpetitia, Convex functions and the Hadamard inequality, Rev. Colombiana Mat., 28 (1994), 7-12.
[5]	J. D. Bukweli-Kyemba, M. N. Hounkonnou, Quantum deformed algebras: Coherent states and special functions, 2013, arXiv:1301.0116v1.
[6]	S. S. Dragomir, R. P. Agarwal, Two inequalities for diferentiable mappings and applications to special means fo real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91-95. doi: 10.1016/S0893-9659(98)00086-X
[7]	T. Ernst, A Comprehensive Treatment of q-Calculus, Springer, Basel, 2012.
[8]	H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl., 47 (2004), 281-300. doi: 10.1016/S0898-1221(04)90025-9
[9]	D. A. Ion, Some estimates on the Hermite-Hadamard inequality through quasi-convex functions, Annals of University of Craiova, Math. Comp. Sci. Ser., 34 (2007), 82-87.
[10]	R. Jagannathan, K. S, Rao, Tow-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series, 2006, arXiv:math/0602613v.
[11]	F. H. Jackson, On a q-definite integrals, Quart. J. Pure Appl. Math., 41 (1910), 193-203.
[12]	M. Kunt, İ. İşcan, N. Alp, et al. (p, q)-Hermite-Hadamard inequalities and (p, q)-estimates for midpoint type inequalities via convex and quasi-convex functions, RACSAM, 112 (2018), 969-992.
[13]	M. Kunt, İ. İşcan, Erratum: Quantum integral inequalities for convex functions, 2016.
[14]	M. Kunt, İ. İşcan, Erratum: Some quantum estimates for Hermite-Hadamard inequalities, 2016.
[15]	V. Kac, P. Cheung, Quantum Calculus, Springer, 2001.
[16]	M. A. Noor, K. I. Noor, M. U. Awan, Some quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675-679.
[17]	M. A. Noor, K. I. Noor, M. U. Awan, Some quantum integral inequalities via preinvex functions, Appl. Math. Comput., 269 (2015), 242-251.
[18]	C. E. M. Pearce, J. Pečarić, Inequalities for diferentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13 (2000), 51-55. doi: 10.1016/S0893-9659(99)00164-0
[19]	A. W. Roberts, D. E. Varberg, Convex functions, Academic Press, New York, 1973.
[20]	P. N. Sadjang, On the fundamental theorem of (p, q)-calculus and some (p, q)-Taylor formulas, 2013, arXiv:1309.3934v1.
[21]	W. Sudsutad, S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, J. Math. Inequal., 9 (2015), 781-793. doi: 10.7153/jmi-09-64
[22]	M. Tunç, E. Göv, (p, q)-Integral inequalities, RGMIA Res. Rep. Coll., 19 (2016), 1-13.
[23]	M. Tunç, E. Göv, Some integral inequalities via (p, q)-calculus on finite intervals, RGMIA Res. Rep. Coll., 19 (2016), 1-12.
[24]	M. Tunç, E. Göv, S. Balgeçti, Simpson type quantum integral inequalities for convex functions, Miskolc Math. Notes, 19 (2018), 649-664. doi: 10.18514/MMN.2018.1661
[25]	J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 121 (2014), 1-13.
[26]	J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 282 (2013), 1-19.
[27]	L. Yang, R. Yang, Some new Hermite-Hadamard type inequalities for h-convex functions via quantum integral on finite intervals, J. Math. Comput. Sci., 18 (2018), 74-86. doi: 10.22436/jmcs.018.01.08
[28]	Y. Zhang, T. S. Du, H. Wang, et al. Different types of quantum integral inequalities via (α, m)- convexity, J. Inequal. Appl., 264 (2018), 1-24.
[29]	H. Zhuang, W. Liu, J.Park, Some quantum estimates of Hermite-Hadamard inequalities for quasiconvex functions, Mathematics, 7 (2019), 1-18. doi: 10.3390/math7020152

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)