On generalizations of quantum Simpson's and quantum Newton's inequalities with some parameters

Chanon Promsakon; Muhammad Aamir Ali; Hüseyin Budak; Mujahid Abbas; Faheem Muhammad; Thanin Sitthiwirattham; Chanon Promsakon; Muhammad Aamir Ali; Hüseyin Budak; Mujahid Abbas; Faheem Muhammad; Thanin Sitthiwirattham

doi:10.3934/math.2021807

AIMS Mathematics

2021, Volume 6, Issue 12: 13954-13975. doi: 10.3934/math.2021807

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Research article

On generalizations of quantum Simpson's and quantum Newton's inequalities with some parameters

1.
Department of Mathematics, Faculty of Applied Science, King Mongkut's University of Technology North Bangkok, Bangkok, 10800, Thailand
2.
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China
3.
Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce-TURKEY
4.
Department of Mathematics, Government College University Lahore, Pakistan
5.
Department of Medical research, China Medical University Hospital, China, Medical University, Taichung, Taiwan
6.
Mathematics Department, Faculty of Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand

Received: 19 July 2021 Accepted: 22 September 2021 Published: 28 September 2021
MSC : 26D10, 26D15, 26A51

In this paper, we prove two identities concerning quantum derivatives, quantum integrals, and some parameters. Using the newly proved identities, we prove new Simpson's and Newton's type inequalities for quantum differentiable convex functions with two and three parameters, respectively. We also look at the special cases of our key findings and find some new and old Simpson's type inequalities, Newton's type inequalities, midpoint type inequalities, and trapezoidal type inequalities.
- Simpson's inequalities,
- Newton's inequalities,
- quantum calculus,
- convex functions
Citation: Chanon Promsakon, Muhammad Aamir Ali, Hüseyin Budak, Mujahid Abbas, Faheem Muhammad, Thanin Sitthiwirattham. On generalizations of quantum Simpson's and quantum Newton's inequalities with some parameters[J]. AIMS Mathematics, 2021, 6(12): 13954-13975. doi: 10.3934/math.2021807

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Abstract

In this paper, we prove two identities concerning quantum derivatives, quantum integrals, and some parameters. Using the newly proved identities, we prove new Simpson's and Newton's type inequalities for quantum differentiable convex functions with two and three parameters, respectively. We also look at the special cases of our key findings and find some new and old Simpson's type inequalities, Newton's type inequalities, midpoint type inequalities, and trapezoidal type inequalities.

References

[1]	S. S. Dragomir, R. P. Agarwal, P. Cerone, On Simpson's inequality and applications, J. Inequal. Appl., 5 (2000), 533–579.
[2]	M. Alomari, M. Darus, S. S. Dragomir, New inequalities of Simpson's type for $s$-convex functions with applications, RGMIA Res. Rep. Coll., 12 (2009).
[3]	M. Z. Sarikaya, E. Set, M. E. Özdemir, On new inequalities of Simpson's type for convex functions, RGMIA Res. Rep. Coll., $\bf13$ (2010).
[4]	M. A. Noor, K. I. Noor, S. Iftikhar, Some Newton's type inequalities for harmonic convex functions, J. Adv. Math. Stud., 9 (2016), 7–16.
[5]	M. A. Noor, K. I. Noor, S. Iftikhar, Newton inequalities for $p$-harmonic convex functions, Honam Math. J., 40 (2018), 239–250.
[6]	S. Iftikhar, S. Erden, P. Kumam, M. U. Awan, Local fractional Newton's inequalities involving generalized harmonic convex functions, Adv. Differ. Equ., 2020 (2020), 1–14. doi: 10.1186/s13662-019-2438-0
[7]	T. A. Ernst, Comprehensive Treatment of $q$ -Calculus, Basel: Springer, 2012.
[8]	V. Kac, P, Cheung, Quantum calculus, New York: Springer, 2002.
[9]	F. Benatti, M. Fannes, R. Floreanini, D. Petritis, Quantum Information, Computation and Cryptography: An Introductory Survey of Theory, Technology and Experiments, Berlin/Heidelberg: Springer Science and Business Media, 2010.
[10]	A. Bokulich, G. Jaeger, Philosophy of Quantum Information Theory and Entaglement, Cambridge: Cambridge Uniersity Press, 2010.
[11]	F. H. Jackson, On a $q$-definite integrals, Quarterly J. Pure Appl. Math., 41 (1910), 193–203.
[12]	T. Ernst, The History Of $Q$-Calculus Furthermore, New Method, Department of Mathematics, Uppsala University: Uppsala, Sweden, 2000.
[13]	W. Al-Salam, Some fractional $q$-integrals and $q$ -derivatives, Proc. Edinb. Math. Soc., 15 (1966), 135–140. doi: 10.1017/S0013091500011469
[14]	J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013 (2013), 282. doi: 10.1186/1687-1847-2013-282
[15]	S. Bermudo, P. Kórus, J. N. Valdés, On $q$ -Hermite-Hadamard inequalities for general convex functions, Acta Math. Hung., 162 (2020), 364–374. doi: 10.1007/s10474-020-01025-6
[16]	P. N. Sadjang, On the fundamental theorem of $(p, q) $ -calculus and some $(p, q)$-Taylor formulas, arXiv 2013, arXiv: 1309.3934v1.
[17]	J. Soontharanon, T. Sitthiwirattham, On fractional $(p, q)$-calculus, Adv. Differ. Equ., 2020 (2020), 1–18.
[18]	M. Tunç, E. Göv, S. Balgeçti, Simposn type quantum integral inequalities for convex functions, Miskolc Math. Notes, 19 (2018), 649–664. doi: 10.18514/MMN.2018.1661
[19]	Y. M. Chu, M. U. Awan, S. Talib, M. A. Noor, K. I. Noor, New post quantum analogues of Ostrowski-type inequalities using new definitions of left–right $\left(p, q\right) $-derivatives and definite integrals, Adv. Differ. Equ., 2020 (2020), 634. doi: 10.1186/s13662-020-03094-x
[20]	M. A. Ali, H. Budak, M. Abbas, Y. M. Chu, Quantum Hermite–Hadamard-type inequalities for functions with convex absolute values of second $q^{b}$-derivatives, Adv. Differ. Equ., 2021 (2021).
[21]	M. A. Ali, N. Alp, H. Budak, Y. M. Chu, Z. Zhang, On some new quantum midpoint type inequalities for twice quantum differentiable convex functions, Open Math., 2021, in press.
[22]	N. Alp, M. Z. Sarikaya, M. Kunt, İ. İşcan, $q$ -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
[23]	N. Alp, M. Z. Sarikaya, Hermite Hadamard's type inequalities for co-ordinated convex functions on quantum integral, Appl. Math. E-Notes, 20 (2020), 341–356.
[24]	H. Budak, Some trapezoid and midpoint type inequalities for newly defined quantum integrals, Proyecciones, 40 (2021), 199–215. doi: 10.22199/issn.0717-6279-2021-01-0013
[25]	H. Budak, M. A. Ali, M. Tarhanaci, Some new quantum Hermite-Hadamard-like inequalities for coordinated convex functions, J. Optim. Theory Appl., 186 (2020), 899–910. doi: 10.1007/s10957-020-01726-6
[26]	S. Jhanthanam, J. Tariboon, S. K. Ntouyas, K. Nonlaopon, On $q$-Hermite-Hadamard inequalities for differentiable convex functions, Mathematics, 7 (2019), 632. doi: 10.3390/math7070632
[27]	M. A. Noor, K. I. Noor, M. U. Awan, Some quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675–679.
[28]	M. A. Noor, K. I. Noor, M. U. Awan, Some quantum integral inequalities via preinvex functions, Appl. Math. Comput., 269 (2015), 242–251.
[29]	E. R. Nwaeze, A. M. Tameru, New parameterized quantum integral inequalities via $\eta $-quasiconvexity, Adv. Differ. Equ., 2019 (2019).
[30]	M. A. Khan, M. Noor, E. R. Nwaeze, Y. M. Chu, Quantum Hermite–Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020 (2020).
[31]	H. Budak, S. Erden, M. A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Meth. Appl. Sci., 44 (2020), 378–390.
[32]	M. A. Ali, H. Budak, Z. Zhang, H. Yildrim, Some new Simpson's type inequalities for co-ordinated convex functions in quantum calculus, Math. Methods Appl. Sci., 44 (2021), 4515–4540. doi: 10.1002/mma.7048
[33]	M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza, Y. M. Chu, New quantum boundaries for quantum Simpson's and quantum Newton's type inequalities for preinvex functions, Adv. Differ. Equ., 2021 (2021), 64. doi: 10.1186/s13662-021-03226-x
[34]	M. Vivas-Cortez, M. A. Ali, A. Kashuri, I. B. Sial, Z. Zhang, Some New Newton's Type Integral Inequalities for Co-Ordinated Convex Functions in Quantum Calculus, Symmetry, 12 (2020), 1476. doi: 10.3390/sym12091476
[35]	M. A. Ali, Y. M. Chu, H. Budak, A. Akkurt, H. Yildrim, Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables, Adv. Differ. Equ., 2021 (2021), 25. doi: 10.1186/s13662-020-03195-7
[36]	M. A. Ali, H. Budak, A. Akkurt, Y. M. Chu, Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus, Open Math., 2021, in press.
[37]	H. Budak, M. A. Ali, T. Tunç, Quantum Ostrowski type integral inequalities for functions of two variables, Math. Meth. Appl. Sci., 44 (2021), 5857–5872. doi: 10.1002/mma.7153
[38]	H. Budak, M. A. Ali, N. Alp, Y. M. Chu, Quantum Ostrowski type integral inequalities, J. Math. Inequal., 2021, in press.
[39]	M. Kunt, İ. İşcan, N. Alp, M. Z. Sarikaya, $ \left(p, q\right) -$Hermite-Hadamard inequalities and $\left(p, q\right) -$ estimates for midpoint inequalities via convex quasi-convex functions, Racsam. Rev. R. Acad. A., 2018 (112), 969–992.
[40]	M. A. Latif, M. Kunt, S. S. Dragomir, İ. İşcan, Post-quantum trapezoid type inequalities, AIMS Math., 2020 (2020), 4011.
[41]	S. Iftikhar, S. Erden, N. Alp, On Generalizations of some inequalities for convex functions via quantum integrals, Submitted, 2020.
[42]	W. Sudsutad, S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, J. Math. Inequal., 9 (2015), 781–793.
[43]	T. Du, Y. Li, Z. Yang, A generalization of Simpson's inequality via differentiable mapping using extended $(s, m)$-convex functions, Appl. Math. Comput., 293 (2017), 358–369.
[44]	S. Erden, S. Iftikhar, N. Alp, Simpson second type estimations for convex functions via quantum calculus, Submitted, 2020.

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