Parametric generalized $ (p, q) $-integral inequalities and applications

Kamsing Nonlaopon; Muhammad Uzair Awan; Sadia Talib; Hüseyin Budak; Kamsing Nonlaopon; Muhammad Uzair Awan; Sadia Talib; Hüseyin Budak

doi:10.3934/math.2022690

AIMS Mathematics

2022, Volume 7, Issue 7: 12437-12457. doi: 10.3934/math.2022690

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Research article

Parametric generalized $ (p, q) $-integral inequalities and applications

1.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Turkey

Received: 10 February 2022 Revised: 24 March 2022 Accepted: 31 March 2022 Published: 26 April 2022
MSC : 05A33, 26A51, 26D10, 26D15

A new generalized $ (p, q) $-integral identity is derived. Using this new identity as an auxiliary result, we derive new parametric generalizations of certain integral inequalities using the class of $ s $-preinvex functions. We discuss several new and known special cases of the obtained results. This shows that our results are quite unifying. To demonstrate the significance of the main results, we also present some interesting applications.
- convex,
- preinvex,
- $ s $-preinvex,
- post quantum calculus,
- integral inequalities
Citation: Kamsing Nonlaopon, Muhammad Uzair Awan, Sadia Talib, Hüseyin Budak. Parametric generalized $ (p, q) $-integral inequalities and applications[J]. AIMS Mathematics, 2022, 7(7): 12437-12457. doi: 10.3934/math.2022690

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Abstract

A new generalized $ (p, q) $-integral identity is derived. Using this new identity as an auxiliary result, we derive new parametric generalizations of certain integral inequalities using the class of $ s $-preinvex functions. We discuss several new and known special cases of the obtained results. This shows that our results are quite unifying. To demonstrate the significance of the main results, we also present some interesting applications.

References

[1]	G. Cristescu, L. Lupsa, Non–connected convexities and applications, Dordrecht, Holland: Kluwer Academic Publishers, 2002.
[2]	S. S. Dragomir, C. E. M. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Australia: Victoria University, 2000.
[3]	M. Z. Sarikaya, E. Set, M. E. Ozdemir, On new inequalities of Simpson's type for $s$-convex functions, Comput. Math. Appl., 60 (2010), 2191–2199. http://dx.doi.org/10.1016/j.camwa.2010.07.033 doi: 10.1016/j.camwa.2010.07.033
[4]	S. S. Dragomir, R. P. Agarwal, P. Cerone, On Simpson's inequality and applications, J. Ineqal. Appl., 5 (2000), 533–579.
[5]	J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013 (2013), 282. http://dx.doi.org/10.1186/1687-1847-2013-282 doi: 10.1186/1687-1847-2013-282
[6]	R. Chakrabarti, R. Jagannathan, A $(p, q)$-oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991), L711–L718. https://doi.org/10.1088/0305-4470/24/13/002 doi: 10.1088/0305-4470/24/13/002
[7]	J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 121. http://dx.doi.org/10.1186/1029-242X-2014-121 doi: 10.1186/1029-242X-2014-121
[8]	W. Sudsutad, S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, J. Math. Inequal., 9 (2015), 781–793. http://dx.doi.org/10.7153/jmi-09-64 doi: 10.7153/jmi-09-64
[9]	M. A. Noor, K. I. Noor, M. U. Awan, Some quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675–679. http://dx.doi.org/10.1016/j.amc.2014.11.090 doi: 10.1016/j.amc.2014.11.090
[10]	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. http://dx.doi.org/10.1016/j.jksus.2016.09.007 doi: 10.1016/j.jksus.2016.09.007
[11]	M. A. Noor, K. I. Noor, M. U. Awan, Some quantum integral inequalities via preinvex functions, Appl. Math. Comput., 269 (2015), 242–251. http://dx.doi.org/10.1016/j.amc.2015.07.078 doi: 10.1016/j.amc.2015.07.078
[12]	M. A. Noor, M. U. Awan, K. I. Noor, Quantum Ostrowski inequalities for $q$–differentiable convex functions, J. Math. Inequal., 10 (2016), 1013–1018. http://dx.doi.org/10.7153/jmi-10-81 doi: 10.7153/jmi-10-81
[13]	Y. Zhang, T. S. Du, H. Wang, Y.-J. Shen, Different types of quantum integral inequalities via $(\alpha, m)$-convexity, J. Inequal. Appl., 2018 (2018), 264. http://dx.doi.org/10.1186/s13660-018-1860-2 doi: 10.1186/s13660-018-1860-2
[14]	T. S. Du, C. Luo, B. Yu, Certain quantum estimates on the parameterized integral inequalities and their applications, J. Math. Inequal., 15 (2021), 201–228. http://dx.doi.org/10.7153/jmi-2021-15-16 doi: 10.7153/jmi-2021-15-16
[15]	Y. P. Deng, M. U. Awan, S. H. Wu, Quantum integral inequalities of Simpson-type for strongly preinvex functions, Mathematics, 7 (2019), 751. http://dx.doi.org/10.3390/math7080751 doi: 10.3390/math7080751
[16]	M. Kunt, I. Iscan, N. Alp, M. Z. Sarikaya, $(p, q)$-Hermite-Hadamard inequalities and $(p, q)$-estimates for midpoint type inequalities via convex and quasi-convex functions, RACSAM, 112 (2018), 969–992. http://dx.doi.org/10.1007/s13398-017-0402-y doi: 10.1007/s13398-017-0402-y
[17]	M. U. Awan, S. Talib, M. A. Noor, Y.-M. Chu, K. I. Noor, On post quantum estimates of upper bounds involving twice $(p, q)$-differentiable preinvex function, J. Inequal. Appl., 2020 (2020), 229. http://dx.doi.org/10.1186/s13660-020-02496-5 doi: 10.1186/s13660-020-02496-5
[18]	Y. F. Tian, Z. S. Wang, A new multiple integral inequality and its application to stability analysis of time–delay systems, Appl. Math. Lett., 105, (2020), 106325. http://dx.doi.org/10.1016/j.aml.2020.106325
[19]	Y. F. Tian, Z. S. Wang, Composite slack–matrix–based integral inequality and its application to stability analysis of time–delay systems, Appl. Math. Lett., 120 (2021), 107252. http://dx.doi.org/10.1016/j.aml.2021.107252 doi: 10.1016/j.aml.2021.107252
[20]	T. Weir, B. Mond, Preinvex functions in multiple objective optimization, J. Math. Anal. Appl., 136 (1988), 29–38. http://dx.doi.org/10.1016/0022-247X(88)90113-8 doi: 10.1016/0022-247X(88)90113-8
[21]	A. Ben-Israel, B. Mond, What is invexity?, The ANZIAM Journal, 28 (1986), 1–9. http://dx.doi.org/10.1017/S0334270000005142
[22]	M. A. Noor, K. I. Noor, M. U. Awan, J. Li, On Hermite-Hadamard inequalities for $h$-preinvex functions, Filomat, 28 (2014), 1463–1474. http://dx.doi.org/10.2298/FIL1407463N doi: 10.2298/FIL1407463N
[23]	S. R. Mohan, S. K. Neogy, On invex sets and preinvex functions, J. Math. Anal. Appl., 189 (1995), 901–908. http://dx.doi.org/10.1006/jmaa.1995.1057 doi: 10.1006/jmaa.1995.1057
[24]	M. Tunc, E. Gov, Some integral inequalities via $(p, q)$-calculus on finite intervals, Filomat, 35 (2021), 1421–1430. http://dx.doi.org/10.2298/FIL2105421T doi: 10.2298/FIL2105421T
[25]	C. P. Niculescu, L.-E. Persson, Convex functions and their applications. A contemporary approach, 2 Eds., Cham: Springer, 2018. http://dx.doi.org/10.1007/978-3-319-78337-6

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