Korean J. Math.  Vol 29, No 1 (2021)  pp.193-203
DOI: https://doi.org/10.11568/kjm.2021.29.1.193

Simpson's and Newton's type quantum integral inequalities for preinvex functions

Muhammad AAmir Ali, Mujahid Abbas, Mubarra Sehar, Ghulam Murtaza

Abstract


In this research, we offer two new quantum integral equalities for recently defined $q^{\varepsilon _{2}}$-integral and derivative, the derived equalities then used to prove quantum integral inequalities of Simpson's and Newton's type for preinvex functions. We also considered the special cases of established results and offer several new and existing results inside the literature of Simpson's and Newton's type inequalities.


Keywords


Simpson's inequalities; q-integral; q-derivative; preinvex function

Subject classification

26D15; 26D10; 26A51

Sponsor(s)

National Natural Science Foundation of China

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References


M. A. Ali, H. Budak, Z. Zhang, and H. Yildrim, Some new Simpson’s type inequalities for co- ordinated convex functions in quantum calculus, Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.7048. (Google Scholar)

M. A. Ali, H. Budak, M. Abbas, and Y.-M. Chu, Quantum Hermite.Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives, Adv Differ Equ 2021 (7) (2021). https://doi.org/10.1186/s13662- 020-03163-1. (Google Scholar)

M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza and Yu-Ming Chu, New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex func- tions, Adv Differ Equ 2021, 64 (2021). (Google Scholar)

https://doi.org/10.1186/s13662-021-03226-x. (Google Scholar)

M. A. Ali, Y.-M. Chu, H. Budak, A. Akkurt, and H.Yildrim, Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables, Adv Differ Equ 2021, 25 (2021). https://doi.org/10.1186/s13662-020-03195-7. (Google Scholar)

M. A. Ali, N. Alp, H. Budak, Y-M. Chu and Z. Zhang, On some new quantum midpoint type inequalities for twice quantum differentiable convex functions, Open Mathematics 2021, in press. (Google Scholar)

M. A. Ali, H. Budak, A. Akkurt and Y-M. Chu, Quantum Ostrowski type inequalities for twice quantum differentiable functions in quantum calculus, Open Mathematics 2021, in press. (Google Scholar)

R. P. Agarwal, A propos d’une note de m. pierre humbert, CR Acad. Sci. Paris 236 (21) (1953), 2031–2032. (Google Scholar)

W. A. Al-Salam, Some fractional q-integrals and q-derivatives, Proceedings of the Edinburgh Mathematical Society, 15 (2) (1966), 135–140. (Google Scholar)

M. Alomari, M. Darus, and S. S. Dragomir, New inequalities of Simpson’s type for s-convex functions with applications, Research report collection, 12 (4) (2009). (Google Scholar)

N. Alp, M. Z. Sarikaya, M. Kunt, and I. Iscan, q- Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, Journal of King Saud University-Science 30 (2) (2018), 193–203. (Google Scholar)

S. Bermudo, P. Korus and J. E. N. Valdes, On q-Hermite Hadamard inequalities for general convex functions, Acta Mathematica Hungarica, pages 1–11, 2020. (Google Scholar)

B. B.-Mohsin, M. U. Awan, M. A. Noor, L. Riahi, K. I. Noor, and B. Almutairi, New quan- tum Hermite-Hadamard inequalities utilizing harmonic convexity of the functions, IEEE Access, 7:20479–20483, 2019. (Google Scholar)

H. Budak, M. A. Ali, M. Tarhanaci, Some new quantum Hermite Hadamard-like inequalities for coordinated convex functions, Journal of Optimization Theory and Applications, pages 1–12, 2020. (Google Scholar)

H. Budak, S. Erden, M. A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Mathematical Methods in the Applied Sciences, 2020. (Google Scholar)

Y. Deng, M. U. Awan, and S. Wu, Quantum integral inequalities of Simpson-type for strongly preinvex functions, Mathematics 7 (8) (2019), 751. (Google Scholar)

S. S. Dragomir, R. P. Agarwal, and P. Cerone, On Simpson’s inequality and applications, RGMIA research report collection 2 (3) (1999). (Google Scholar)

T.-S. Du, J.-G. Liao, and Y.-J. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized (s, m)-preinvex functions J. Nonlinear Sci. Appl 9 (5) (2016), 3112–3126. (Google Scholar)

T. Ernst, The history of q-calculus and a new method, Citeseer, Sweden, 2000. (Google Scholar)

T. Ernst, A comprehensive treatment of q-calculus, Springer, Science and Business Media, 2012. (Google Scholar)


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