In this article, we introduce the notion of $ _{a}{\bar{T}}_{p,q} $-integrals. Using the definition of $ _{a}{\bar{T}}_{p,q} $-integrals, we derive some new post quantum analogues of some classical results of Young's inequality, Hölder's inequality, Minkowski's inequality, Ostrowski's inequality and Hermite-Hadamard's inequality.
Citation: Bandar Bin-Mohsin, Muhammad Uzair Awan, Muhammad Zakria Javed, Sadia Talib, Hüseyin Budak, Muhammad Aslam Noor, Khalida Inayat Noor. On some classical integral inequalities in the setting of new post quantum integrals[J]. AIMS Mathematics, 2023, 8(1): 1995-2017. doi: 10.3934/math.2023103
In this article, we introduce the notion of $ _{a}{\bar{T}}_{p,q} $-integrals. Using the definition of $ _{a}{\bar{T}}_{p,q} $-integrals, we derive some new post quantum analogues of some classical results of Young's inequality, Hölder's inequality, Minkowski's inequality, Ostrowski's inequality and Hermite-Hadamard's inequality.
[1] | N. Alp, M. Z. Sarikaya, A new definition and properties of quantum integral which calls $\bar{q}$–integral, Konuralp J. Math., 5 (2017), 146–159. |
[2] | N. Alp, M. Z. Sarikaya, $\overline {q} $-Inequalities on quantum integral, Malaya J. Matematik, 8 (2020), 2035–2044. https://doi.org/10.26637/MJM0804/0121 doi: 10.26637/MJM0804/0121 |
[3] | N. Alp, M. Z. Sarikaya, M. Kunt, I. Iscan, $q$–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. https://doi.org/10.1016/j.jksus.2016.09.007 doi: 10.1016/j.jksus.2016.09.007 |
[4] | S. Bermudo, P. Kórus, J. E. N. Valdés, On ${q}$-Hermite-Hadamard inequalities for general convex functions, Acta Math. Hung., 162 (2020), 364–374. |
[5] | R. Chakrabarti, R. Jagannathan, A $(p, q)$–oscillator realization of two-parameter quantum algebras, J. Phys. A: Math. Gen., 24 (1991), L711. |
[6] | Y. M. Chu, M. U. Awan, S. Talib, M. A. Noor, K. I. Noor, New post quantum analogues of Ostrowski type inequalities using new definitions of left–right $(p,q)$-derivatives and definite integrals, Adv. Differ. Equ., 2020 (2020), 634. https://doi.org/10.1186/s13662-020-03094-x doi: 10.1186/s13662-020-03094-x |
[7] | T. S. Du, C. Y. Luo, B. Yu, Certain quantum estimates on the parameterized integral inequalities and their applications, J. Math. Inequal., 15 (2021), 201–228. https://doi.org/10.7153/jmi-2021-15-16 doi: 10.7153/jmi-2021-15-16 |
[8] | F. H. Jackson, On a ${q}$-definite integrals, Q. J. Pure Appl. Math., 41 (1910), 193–203. |
[9] | V. Kac, P. Cheung, Quantum calculus, Springer, 2002. |
[10] | H. Kara, H. Budak, N. Alp, H. Kalsoom, M. Z. Sarikaya, On new generalized quantum integrals and related Hermite-Hadamard inequalities, J. Inequal. Appl., 2021 (2021), 180. https://doi.org/10.1186/s13660-021-02715-7 doi: 10.1186/s13660-021-02715-7 |
[11] | M. Kunt, I. Işcan, N. Alp, M. Z. Sarikaya, $(p, q)$–Hermite–Hadamard inequalities and $(p,q)$–estimates for midpoint type inequalities via convex and quasi–convex functions, Rascam. Rev. R. Acad. Math., 112 (2018), 969–992. https://doi.org/10.1007/s13398-017-0402-y doi: 10.1007/s13398-017-0402-y |
[12] | M. A. Noor, K. I. Noor, M. U. Awan, Some quantum estimates for Hermite–Hadamard inequalities, Appl. Math. Comput., 251 (2015), 675–679. https://doi.org/10.1016/j.amc.2014.11.090 doi: 10.1016/j.amc.2014.11.090 |
[13] | W. Sudsutad, S. K. Ntouyas, J. Tariboon, Quantum integral inequalities for convex functions, J. Math. Inequal., 9 (2015), 781–793. https://doi.org/10.7153/jmi-09-64 doi: 10.7153/jmi-09-64 |
[14] | J. Tariboon, S. K. Ntouyas, Quantum calculus on finite intervals and applications to impulsive difference equations, Adv. Differ. Equ., 2013 (2013), 282. https://doi.org/10.1186/1687-1847-2013-282 doi: 10.1186/1687-1847-2013-282 |
[15] | J. Tariboon, S. K. Ntouyas, Quantum integral inequalities on finite intervals, J. Inequal. Appl., 2014 (2014), 121. https://doi.org/10.1186/1029-242X-2014-121 doi: 10.1186/1029-242X-2014-121 |
[16] | M. Tunc, E. Gov, Some integral inequalities via $(p,q)$-calculus on finite intervals, Filomat, 35 (2021), 1421–1430. https://doi.org/10.2298/FL2105421T doi: 10.2298/FL2105421T |
[17] | M. Vivas-Cortez, M. A. Ali, H. Budak, H. Kalsoom, P. Agarwal, Some new Hermite–Hadamard and related inequalities for convex functions via $\mathit{(p,q)}$-integral, entropy, Entropy, 23 (2021), 828. https://doi.org/10.3390/e23070828 doi: 10.3390/e23070828 |
[18] | B. Yu, C. Y. Luo, T. S. Du, On the refinements of some important inequalities via $(p, q)$–calculus and their applications, J. Inequal. Appl., 2021 (2021), 1–26. https://doi.org/10.1186/s13660-021-02617-8 doi: 10.1186/s13660-021-02617-8 |
[19] | Y. Zhang, T. S. Du, H. Wang, Y. J. Shen, Different types of quantum integral inequalities via $(\alpha,m)$- convexity, J. Inequal. Appl., 2018 (2018), 1–24. https://doi.org/10.1186/s13660-018-1860-2 doi: 10.1186/s13660-018-1860-2 |
[20] | M. A. Abbas, L. Chen, A. R. Khan, G. Muhammad, B. Sun, S. Hussain, et al., Some new Anderson type $h$ and $q$ integral inequalities in quantum calculus, Symmetry, 14 (2022), 1294. https://doi.org/10.3390/sym14071294 doi: 10.3390/sym14071294 |
[21] | Y. Deng, M. U. Awan, S. Wu, Quantum integral inequalities of Simpson-type for strongly preinvex functions, Mathematics, 7 (2019), 751. https://doi.org/10.3390/math7080751 doi: 10.3390/math7080751 |
[22] | S. Erden, S. Iftikhar, M. R. Delavar, P. Kumam, P. Thounthong, W. Kumam, On generalizations of some inequalities for convex functions via quantum integrals, Rascam. Rev. R. Acad. Math., 114 (2020), 1–15. https://doi.org/10.1007/s13398-020-00841-3 doi: 10.1007/s13398-020-00841-3 |
[23] | D. F. Zhao, G. Gulshan, M. A. Ali, K. Nonlaopon, Some new midpoint and trapezoidal-type inequalities for general convex functions in $q$-calculus, Mathematics, 10 (2022), 444. https://doi.org/10.3390/math10030444 doi: 10.3390/math10030444 |
[24] | B. Bin-Mohsin, M. Saba, M. Z. Javed, M. U. Awan, H. Budak, K. Nonlaopon, A quantum calculus view of Hermite-Hadamard-Jensen-Mercer inequalities with applications, Symmetry, 14 (2022), 1246. https://doi.org/10.3390/sym14061246 doi: 10.3390/sym14061246 |