International Journal of Nonlinear Analysis and Applications

Quantum dual Simpson type inequalities for q-differentiable convex functions

Document Type : Research Paper

Authors

1 Department of Mathematics, Taibah University, Al-Medina 20012, Saudi Arabia

2 Department of Mathematics, 8 may 1945 University, Guelma 24000, Algeria

3 Department CPST, Ecole Nationale Sup´erieure de Technologie et d’Ing´enierie, Annaba, 23005, Algeria

Abstract

This work introduces the quantum analogue of the dual Simpson type integral inequalities for the class of q-differentiable convex functions through a new identity. The results are also accompanied by their applications.

Keywords

[1] M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza and Y.-M. Chu, New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions, Adv. Difference Equ. (2021), Paper No. 64.
[2] M. A. Ali, H. Budak, Z. Zhang, and H. Yildirim, Some new Simpson’s type inequalities for coordinated convex functions in quantum calculus, Math. Methods Appl. Sci. 44 (2021), no. 6, 4515–4540.
[3] M. A. Ali, H. Budak and Z. Zhang, A new extension of quantum Simpson’s and quantum Newton’s type inequalities for quantum differentiable convex functions, Math. Methods Appl. Sci 45 (2022), no. 4, 1845–1863.
[4] O.B. Almutairi, Quantum estimates for different type intequalities through generalized convexity, Entropy 24 (2022), no. 5, Paper No. 728.
[5] N. Alp, M. Z. Sarikaya, M. Kunt and ˙I. ˙I¸scan, 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.
[6] M.U. Awan, S. Talib, A. Kashuri, M.A. Noor and Y.-M. Chu, Estimates of quantum bounds pertaining to new q-integral identity with applications, Adv. Difference Equ. (2020), Paper No. 424.
[7] H. Budak, S. Erden and M.A. Ali, Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci. 44 (2021), no. 1, 378–390.
[8] Lj. Dedi´c, M. Mati´c and J. Peˇcari´c, On dual Euler-Simpson formulae. Bull. Belg. Math. Soc. Simon Stevin 8 (2001), no. 3, 479–504.
[9] F. H. Jackson, On a q-Definite Integrals, Quart. J. Pure and Appl. Math. 41 (1910) 193–203.
[10] V. Kac and P. Cheung, Quantum calculus. Universitext. Springer-Verlag, New York, 2002.
[11] H. Kalsoom, S. Rashid, M. Idrees, Y.-M. Chu and D. Baleanu, Two-variable quantum integral inequalities of Simpson-type based on higher-order generalized strongly preinvex and quasi-preinvex functions. Symmetry 12 (2019), no. 1, 51.
[12] M. Kunt, ˙I. ˙I¸scan, N. Alp and M.Z. Sarıkaya, (p, q)-Hermite-Hadamard inequalities and (p, q)-estimates for midpoint type inequalities via convex and quasi-convex functions, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat. RACSAM 112 (2018), no. 4, 969–992.
[13] W. Liu and H. Zhuang, Some quantum estimates of Hermite-Hadamard inequalities for convex functions, J. Appl. Anal. Comput. 7 (2017), no. 2, 501–522.
[14] M.A. Noor, K.I. Noor and M.U. Awan, Some quantum estimates for Hermite-Hadamard inequalities, Appl. Math. Comput. 251 (2015), 675–679.
[15] M. A. Noor, M. U. Awan and K. I. Noor, Quantum Ostrowski inequalities for q-differentiable convex functions. J. Math. Inequal. 10 (2016), no. 4, 1013–1018.
[16] M.A. Noor, G. Cristescu and M. U. Awan, Bounds having Riemann type quantum integrals via strongly convex functions, Studia Sci. Math. Hungar. 54 (2017), no. 2, 221–240.
[17] J. E. Peˇcari´c, F. Proschan and Y. L. Tong, Convex functions, partial orderings, and statistical applications, Mathematics in Science and Engineering, 187. Academic Press, Inc., Boston, MA, 1992.
[18] P. Siricharuanun, S. Erden, M.A. Ali, H. Budak, S. Chasreechai and T. Sitthiwirattham, Some new Simpson’s and Newton’s formulas type inequalities for convex functions in quantum calculus, Mathematics 9 (2021), no. 16, 1992.
[19] J. Soontharanon, M.A. Ali, H. Budak, K. Nonlaopon and Z. Abdullah, Simpson’s and Newton’s Type Inequalities for (α, m)-convex functions via quantum calculus, Symmetry 14 (2022), no. 4, 736.
[20] W. Sudsutad, S.K. Ntouyas and J. Tariboon, Quantum integral inequalities for convex functions, J. Math. Inequal. 9 (2015), no. 3, 781–793.
[21] J. Tariboon and S.K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Difference Equ. 2013 (2013), 282.
[22] J. Tariboon and S.K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl. (2014), 2014:121, 13 pp.
[23] M. Tun¸c, E. G¨ov and S. Balge¸cti, Simpson type quantum integral inequalities for convex functions, Miskolc Math. Notes 19 (2018), no. 1, 649–664.
[24] M.J. Vivas-Cortez, M.A. Ali, S. Qaisar, I.B. Sial, S. Jansem and A. Mateen, On some new Simpson’s formula type inequalities for convex functions in post-quantum calculus, Symmetry 13 (2021), no. 12, 2419.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 4
April 2023
Pages 63-76
  • Receive Date: 12 December 2022
  • Revise Date: 11 February 2023
  • Accept Date: 13 February 2023