Quantum analog of some trapezoid and midpoint type inequalities for convex functions

Year 2022, Volume: 71 Issue: 2, 456 - 480, 30.06.2022

Abdul Baidar , Mehmet Kunt

https://doi.org/10.31801/cfsuasmas.1009988

Abstract

In this paper a new quantum analog of Hermite-Hadamard inequality is presented, and based on it, two new quantum trapezoid and midpoint identities are obtained. Moreover, the quantum analog of some trapezoid and midpoint type inequalities are established.

Keywords

Trapezoid inequality, convex function, q-Integral, midpoint inequality

References

• Ali, M. A., Abbas, M., Budak, H., Agarwal, P., Murtaza, G., Chu, Y. M., New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions, Adv. Differ. Equ., 2021(64) (2021), 1-21. https://doi.org/10.1186/s13662-021-03226-x
• Ali, M. A., Alp, N., Budak, H., Chu, Y. M., Zhang, Z., On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions, Open Math., 19(1) (2021), 427-439. https://doi.org/10.1515/math-2021-0015
• Ali, M. A., Budak, H., Abbas, M., Chu, Y. M., Quantum Hermite-Hadamard-type inequalities for functions with convex absolute values of second qb-derivatives, Adv. Differ. Equ., 2021(7) (2021), 1-12. https://doi.org/10.1186/s13662-020-03163-1
• Ali, M. A., Budak, H., Akkurt, A., Chu, Y. M., Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus, Open Math., 19(1) (2021), 440-449. https://doi.org/10.1515/math-2021-0020
• Ali, M. A., Budak, H., Zhang, Z., Yildirim, H., Some new Simpson’s type inequalities for coordinated convex functions in quantum calculus, Math. Methods Appl. Sci., 44(6) (2021), 4515-4540. https://doi.org/10.1002/mma.7048
• Ali, M. A., Chu, Y. M., Budak, H., Akkurt, A., Yildirim, H., Zahid, M. A., Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables, Adv. Differ. Equ., 2021(25) (2021), 1-26. https://doi.org/10.1186/s13662-020-03195-7
• Annaby, M. H., Mansour, Z. S., q-Fractional Calculus and Equations, Springer, Heidelberg, 2012.
• Alp, N., Sarikaya, M. Z., Kunt, M., Iscan, I., q-Hermite Hadamard inequalities and quantum estimates for midpoint type inequalities via convex and quasi-convex functions, J. King Saud Univ.-Sci., 30(2) (2018), 193-203. https://doi.org/10.1016/j.jksus.2016.09.007
• Awan, M. U., Talib, S., Kashuri, A., Noor, M. A., Chu, Y. M., Estimates of quantum bounds pertaining to new q-integral identity with applications, Adv. Differ. Equ., 2020(424) (2020), 1-15. https://doi.org/10.1186/s13662-020-02878-5
• Awan, M. U., Talib, S., Kashuri, A., Noor, M. A., Noor, K. I., Chu, Y. M., A new q-integral identity and estimation of its bounds involving generalized exponentially μ-preinvex functions, Adv. Differ. Equ., 2020(575) (2020), 1-12. https://doi.org/10.1186/s13662-020-03036-7
• Awan, M. U., Talib, S., Noor, M. A., Noor, K. I., Chu, Y. M, On post quantum integral inequalities, J. Math. Inequal., 15(2) (2021), 629-654. https://doi.org/10.7153/jmi-2021-15- 46
• Budak, H., Ali, M. A., Tarhanaci, M., Some new quantum Hermite–Hadamard-like inequalities for coordinated convex functions, J. Optim. Theory. Appl., 186(3) (2020), 899-910. https://doi.org/10.1007/s10957-020-01726-6
• Budak, H., Ali, M. A., Tunc, T., Quantum Ostrowski-type integral inequalities for functions of two variables, Math. Methods Appl. Sci., 44(7) (2021), 5857-5872. https://doi.org/10.1002/mma.7153
• Budak, H., Erden, S., Ali, M. A., Simpson and Newton type inequalities for convex functions via newly defined quantum integrals, Math. Methods Appl. Sci., 44(1) (2021), 378-390. https://doi.org/10.1002/mma.6742
• Budak, H., Khan, S., Ali, M. A., Chu, Y. M., Refinements of quantum Hermite-Hadamardtype inequalities, Open Math., 19(1) (2021), 724-734. https://doi.org/10.1515/math-2021-0029
• Dragomir, S. S., Agarwal, R., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11(5) (1998), 91-95. https://doi.org/10.1016/S0893-9659(98)00086-X
• Du, T. S., Luo, C. Y., Yu, B., Certain quantum estimates on the parameterized integral inequalities and their applications, J. Math. Inequal., 15(1) (2021), 201-228. https://doi.org/10.7153/jmi-2021-15-16
• Eftekhari, N., Some remarks on (s, m)-convexity in the second sense, J. Math. Inequal., 8(3) (2014), 489-495. https://doi.org/10.7153/jmi-08-36
• Erden, S., Iftikhar, S., Delavar, M. R., Kumam, P., Thounthong, P., Kumam, W., On generalizations of some inequalities for convex functions via quantum integrals, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(110) (2020), 1-15. https://doi.org/10.1007/s13398-020-00841-3
• Kac, V., Pokman C., Quantum Calculus, Springer, New York, 2001.
• Kavurmaci, H., Avci, M., Ozdemir, M. E., New inequalities of Hermite-Hadamard type for convex functions with applications, J. Inequal.Appl., 2011(86) (2011), 1-11. https://doi.org/10.1186/1029-242X-2011-86
• Kirmaci, U. S., Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput., 147(1) (2004), 137-146. https://doi.org/10.1016/S0096-3003(02)00657-4
• Kunt, M., Baidar, A., Sanli, Z., Left-Right quantum derivatives and definite integrals, (2020), https://www.researchgate.net/publication/343213377 (Preprint).
• Li, Y. X., Ali, M. A., Budak, H., Abbas, M., Chu, Y. M., A new generalization of some quantum integral inequalities for quantum differentiable convex functions, Adv. Differ. Equ., 2021(225) (2021), 1-15. https://doi.org/10.1186/s13662-021-03382-0
• Khan, M. A., Mohammad, N., Nwaeze, E. R., Chu, Y. M., Quantum Hermite-Hadamard inequality by means of a Green function, Adv. Differ. Equ., 2020(99) (2020), 1-20. https://doi.org/10.1186/s13662-020-02559-3
• Noor, M. A., Noor, K. I., Awan, M. U., Some quantum estimates for Hermite–Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675-679. https://doi.org/10.1016/j.amc.2014.11.090
• Prabseang, J., Nonlaopon, K., Ntouyas, S. K., On the refinement of quantum Hermite-Hadamard inequalities for continuous convex functions, J. Math. Inequal., 14(3) (2020), 875-885. https://doi.org/10.7153/jmi-2020-14-57
• Pearce, C. E. M., Pecaric, J., Inequalities for differentiable mappings with application to special means and quadrature formulae, Appl. Math. Lett., 13(2) (2000), 51-55. https://doi.org/10.1016/S0893-9659(99)00164-0
• Rashid, S., Butt, S. I., Kanwal, S., Ahmad, H., Wang, M. K., Quantum integral inequalities with respect to Raina’s function via coordinated generalized-convex functions with applications, J. Funct. Spaces, Article ID 6631474 (2021). https://doi.org/10.1155/2021/6631474
• Sudsutad, W., Ntouyas, S. K., Tariboon, J., Quantum integral inequalities for convex functions, J. Math. Inequal., 9(3) (2015), 781-793. https://doi.org/10.7153/jmi-09-64
• Tariboon, J., Ntouyas, S. K., Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013(282) (2013), 1-19. https://doi.org/10.1186/1687-1847-2013-282
• Vivas-Cortez, M., Ali, M. A., Kashuri, A., Sial, B. I., Zhang, Z., Some new Newton’s type integral inequalities for co-ordinated convex functions in quantum calculus, Symmetry, 12(9) (2020), 1-28. https://doi.org/10.3390/sym12091476
• Vivas-Cortez, M., Kashuri, A., Liko, R., Hern´andez, J. E. H., Some new q-integral inequalities using generalized quantum Montgomery identity via preinvex functions, Symmetry, 12(4) (2020), 1-15. https://doi.org/10.3390/sym12040553
• You, X., Ali, M. A., Erden, S., Budak, H., Chu, Y. M., On some new midpoint inequalities for the functions of two variables via quantum calculus, J. Inequal. Appl., 2021(142) (2021), 1-23. https://doi.org/10.1186/s13660-021-02678-9
• You, X., Kara, H., Budak, H., Kalsoom, H., Quantum inequalities of Hermite–Hadamard type for-convex functions, J. Math., Article ID 6634614 (2021). https://doi.org/10.1155/2021/6634614
• Zhou, S. S., Rashid, S., Noor, M. A., Noor, K. I., Safdar, F., Chu, Y. M., New Hermite-Hadamard type inequalities for exponentially convex functions and applications, AIMS Math., 5(6) (2020), 6874-6901. https://doi.org/10.3934/math.2020441