Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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SIMPSON'S AND NEWTON'S TYPE QUANTUM INTEGRAL INEQUALITIES FOR PREINVEX FUNCTIONS

  • Ali, Muhammad Aamir (Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University) ;
  • Abbas, Mujahid (Department of Mathematics, Government College University) ;
  • Sehar, Mubarra (Department of Mathematics, Government College University) ;
  • Murtaza, Ghulam (Department of Mathematics (SSC) University of Management and Technology)
  • Received : 2021.02.16
  • Accepted : 2021.03.22
  • Published : 2021.03.30
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Abstract

In this research, we offer two new quantum integral equalities for recently defined qε2-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.

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References

  1. M. A. Ali, H. Budak, Z. Zhang, and H. Yildrim, Some new Simpson's type inequalities for coordinated convex functions in quantum calculus, Mathematical Methods in the Applied Sciences, https://doi.org/10.1002/mma.7048.
  2. 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.
  3. 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 functions, Adv Differ Equ 2021, 64 (2021). https://doi.org/10.1186/s13662-021-03226-x.
  4. 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.
  5. 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.
  6. 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.
  7. R. P. Agarwal, A propos d'une note de m. pierre humbert, CR Acad. Sci. Paris 236 (21) (1953), 2031-2032.
  8. W. A. Al-Salam, Some fractional q-integrals and q-derivatives, Proceedings of the Edinburgh Mathematical Society, 15 (2) (1966), 135-140. https://doi.org/10.1017/S0013091500011469
  9. 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).
  10. 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. https://doi.org/10.1016/j.jksus.2016.09.007
  11. 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.
  12. B. B.-Mohsin, M. U. Awan, M. A. Noor, L. Riahi, K. I. Noor, and B. Almutairi, New quantum Hermite-Hadamard inequalities utilizing harmonic convexity of the functions, IEEE Access, 7:20479-20483, 2019. https://doi.org/10.1109/access.2019.2897680
  13. 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.
  14. 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.
  15. Y. Deng, M. U. Awan, and S. Wu, Quantum integral inequalities of Simpson-type for strongly preinvex functions, Mathematics 7 (8) (2019), 751. https://doi.org/10.3390/math7080751
  16. S. S. Dragomir, R. P. Agarwal, and P. Cerone, On Simpson's inequality and applications, RGMIA research report collection 2 (3) (1999).
  17. 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. https://doi.org/10.22436/jnsa.009.05.102
  18. T. Ernst, The history of q-calculus and a new method, Citeseer, Sweden, 2000.
  19. T. Ernst, A comprehensive treatment of q-calculus, Springer, Science and Business Media, 2012.
  20. H. Gauchman, Integral inequalities in q-calculus, Computers and Mathematics with Applications 47 (2-3) (2004), 281-300, 2004. https://doi.org/10.1016/S0898-1221(04)90025-9
  21. S. Iftikhar, P. Kumam, and S. Erden, Newton's-type integral inequalities via local fractional integrals, Fract 28 (3)(2020), 2050037.604.
  22. D. O. Jackson and T. Fukuda, O. Dunn, and E. Majors, On q-definite integrals, In Quart. J. Pure Appl. Math. Citeseer, 1910.
  23. S. Jhanthanam, J. Tariboon, S. K Ntouyas, and K. Nonlaopon, On q-Hermite-Hadamard inequalities for differentiable convex functions, Mathematics 7 (7) (2019), 632. https://doi.org/10.3390/math7070632
  24. V. Kac and P. Cheung, Quantum calculus, Springer, Science and Business Media, 2001.
  25. M. A. Khan, N. Mohammad, E. R. Nwaeze, and Y.-M. Chu, Quantum Hermite.Hadamard inequality by means of a green function, Advances in Difference Equations 2020 (1) (2020), 1-20. https://doi.org/10.1186/s13662-019-2438-0
  26. W. Liu and H. Zhuang, Some quantum estimates of Hermite-Hadamard inequalities for convex functions, 2016.
  27. M. A. Noor, K. I. Noor, and S. Iftikhar, Some Newton.s type inequalities for harmonic convex functions, J. Adv. Math. Stud 9 (1) (2016), 7-16.
  28. M. A. Noor, M. U. Awan, and K. I. Noor, Quantum ostrowski inequalities for q-differentiable convex functions, J. Math. Inequal 10 (4) (2016), 1013-1018. https://doi.org/10.7153/jmi-10-81
  29. M. A. Noor, K. I. Noor, and M. U. Awan, Some quantum estimates for Hermite- Hadamard inequalities, Applied Mathematics and Computation 251 :675.679, (2015). https://doi.org/10.1016/j.amc.2014.11.090
  30. M. A. Noor, K. I. Noor, and M. U. Awan, Some quantum integral inequalities via preinvex functions, Applied Mathematics and Computation, 269:242.251, 2015. https://doi.org/10.1016/j.amc.2015.07.078
  31. M. A. Noor, K. I. Noor, and S. Iftikhar, Newton inequalities for p-harmonic convex functions, Honam Mathematical Journal 40 (2) (2018), 239-250. https://doi.org/10.5831/HMJ.2018.40.2.239
  32. E. R. Nwaeze and A. M. Tameru, New parameterized quantum integral inequalities via eta-quasiconvexity Advances in Difference Equations 2019 (1) (2019), 425. https://doi.org/10.1186/s13662-019-2358-z
  33. M. Z. Sarikaya, E. Set, and M. E. Ozdemir, On new inequalities of Simpson's type for s-convex functions, Computers and Mathematics with Applications 60 (8) (2010), 2191-2199. https://doi.org/10.1016/j.camwa.2010.07.033
  34. J. Tariboon and S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Advances in Difference Equations 2013 (1) (2013), 282. https://doi.org/10.1186/1687-1847-2013-282
  35. M. Tunc, E. Gov, and S. Balgecti, Simpson type quantum integral inequalities for convex functions, Miskolc Math. Notes 19 (1) (2018), 649-664. https://doi.org/10.18514/mmn.2018.1661