Research Article
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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

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

Abdul Baidar Mehmet Kunt

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

Abstract

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
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences, Applied Mathematics
Journal Section Research Articles
Authors

Abdul Baidar 0000-0002-9510-3138

Mehmet Kunt 0000-0002-8730-5370

Publication Date June 30, 2022
Submission Date October 15, 2021
Acceptance Date December 9, 2021
Published in Issue Year 2022 Volume: 71 Issue: 2

Cite

APA Baidar, A., & Kunt, M. (2022). Quantum analog of some trapezoid and midpoint type inequalities for convex functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(2), 456-480. https://doi.org/10.31801/cfsuasmas.1009988
AMA Baidar A, Kunt M. Quantum analog of some trapezoid and midpoint type inequalities for convex functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2022;71(2):456-480. doi:10.31801/cfsuasmas.1009988
Chicago Baidar, Abdul, and Mehmet Kunt. “Quantum Analog of Some Trapezoid and Midpoint Type Inequalities for Convex Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71, no. 2 (June 2022): 456-80. https://doi.org/10.31801/cfsuasmas.1009988.
EndNote Baidar A, Kunt M (June 1, 2022) Quantum analog of some trapezoid and midpoint type inequalities for convex functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71 2 456–480.
IEEE A. Baidar and M. Kunt, “Quantum analog of some trapezoid and midpoint type inequalities for convex functions”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 71, no. 2, pp. 456–480, 2022, doi: 10.31801/cfsuasmas.1009988.
ISNAD Baidar, Abdul - Kunt, Mehmet. “Quantum Analog of Some Trapezoid and Midpoint Type Inequalities for Convex Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 71/2 (June 2022), 456-480. https://doi.org/10.31801/cfsuasmas.1009988.
JAMA Baidar A, Kunt M. Quantum analog of some trapezoid and midpoint type inequalities for convex functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71:456–480.
MLA Baidar, Abdul and Mehmet Kunt. “Quantum Analog of Some Trapezoid and Midpoint Type Inequalities for Convex Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 71, no. 2, 2022, pp. 456-80, doi:10.31801/cfsuasmas.1009988.
Vancouver Baidar A, Kunt M. Quantum analog of some trapezoid and midpoint type inequalities for convex functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2022;71(2):456-80.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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