Quantum trapezoid-type inequalities for coordinated strongly s-convex functions

Chanokgan Sahatsathatsana; Sattra Sahatsathatsana

doi:10.46481/jnsps.2027.3670

Authors

Chanokgan Sahatsathatsana chanokgan.na@ksu.ac.th Department of Science and Mathematics, Faculty of Science and Health Technology, Kalasin University, Kalasin, Thailand
Sattra Sahatsathatsana Department of English, Faculty of Liberal Arts, Kalasin University, Kalasin, Thailand

Keywords:

Quantum calculus, Trapezoid-type inequalities, Coordinated strong s-convexity, q-integrals, Convexity

Abstract

We establish new q-trapezoid-type inequalities for q-differentiable coordinated strongly s-convex functions on bidimensional intervals. A symmetric modulus convention is adopted for the two coordinates, and the main estimates are derived using a quantum integral identity, the q-Hölder inequality, and the q-power mean inequality. The corresponding classical inequalities are obtained by letting q₁, q₂ → 1⁻. Explicit examples verify the required strong s-convexity assumptions and illustrate the applicability of the results. Numerical computations on a specified finite grid, together with graphical illustrations, support the theoretical estimates.

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References

[1] M. W. Alomari, M. Darus & S. S. Dragomir, ``New inequalities of Hermite--Hadamard type for functions whose second derivatives absolute values are quasi-convex'', Tamkang Journal of Mathematics 41 (2010) 353. https://doi.org/10.5556/j.tkjm.41.2010.498.

[2] S. S. Dragomir & C. E. M. Pearce, Selected topics on Hermite--Hadamard inequalities and applications, RGMIA Monographs, Victoria University, Melbourne, Australia, 2000. https://rgmia.org/monographs/HHCont.htm.

[3] S. S. Dragomir & R. P. Agarwal, ``Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula'', Applied Mathematics Letters 11 (1998) 91. https://doi.org/10.1016/S0893-9659(98)00086-X.

[4] S. S. Dragomir, ``On some new inequalities of Hermite--Hadamard type for $m$-convex functions'', Tamkang Journal of Mathematics 33 (2002) 45. https://doi.org/10.5556/j.tkjm.33.2002.304.

[5] S. Karamardian, ``The nonlinear complementarity problem with applications, part 2'', Journal of Optimization Theory and Applications 4 (1969) 167. https://doi.org/10.1007/BF00930577.

[6] S. S. Dragomir, ``On the Hadamard inequality for convex functions on the coordinates in a rectangle from the plane'', Taiwanese Journal of Mathematics 5 (2001) 775. https://doi.org/10.11650/twjm/1500574995.

[7] M. A. Khan, S. Z. Ullah & Y.-M. Chu, ``The concept of coordinate strongly convex functions and related inequalities'', Revista de la Real Academia de Ciencias Exactas, F'isicas y Naturales. Serie A. Matem'aticas 113 (2019) 2235. https://doi.org/10.1007/s13398-018-0615-8.

[8] J. Bisht, N. Sharma, S. K. Mishra & A. Hamdi, ``Some new integral inequalities for higher-order strongly exponentially convex functions'', Journal of Inequalities and Applications 2023 (2023) 41. https://doi.org/10.1186/s13660-023-02952-y.

[9] K. K. Lai, J. Bisht, N. Sharma & S. K. Mishra, ``Hermite--Hadamard-type integral inequalities for the class of strongly convex functions on time scales'', Journal of Mathematical Inequalities 16 (2022) 975. https://doi.org/10.7153/jmi-2022-16-66.

[10] N. Merentes & K. Nikodem, ``Remarks on strongly convex functions'', Aequationes Mathematicae 80 (2010) 193. https://doi.org/10.1007/s00010-010-0043-0.

[11] S. Z. Ullah, M. A. Khan, Z. A. Khan & Y.-M. Chu, ``Coordinate strongly $s$-convex functions and related results'', Journal of Mathematical Inequalities 14 (2020) 829. https://doi.org/10.7153/jmi-2020-14-53.

[12] F. H. Jackson, ``On $q$-definite integrals'', Quarterly Journal of Pure and Applied Mathematics 41 (1910) 193. https://www.scienceopen.com/document?vid=be95a636-f657-48de-9b1a-68eec3e4fcd6.

[13] R. P. Agarwal, ``Certain fractional $q$-integrals and $q$-derivatives'', Mathematical Proceedings of the Cambridge Philosophical Society 66 (1969) 365. https://doi.org/10.1017/S0305004100045060.

[14] W. A. Al-Salam, ``Some fractional $q$-integrals and $q$-derivatives'', Proceedings of the Edinburgh Mathematical Society 15 (1966) 135. https://doi.org/10.1017/S0013091500011469.

[15] S. Bermudo, P. K'orus & J. N'apoles Vald'es, ``On $q$-Hermite--Hadamard inequalities for general convex functions'', Acta Mathematica Hungarica 162 (2020) 364. https://doi.org/10.1007/s10474-020-01025-6.

[16] J. Tariboon & S. K. Ntouyas, ``Quantum calculus on finite intervals and applications to impulsive difference equations'', Advances in Difference Equations 2013 (2013) 282. https://doi.org/10.1186/1687-1847-2013-282.

[17] S. K. Mishra, R. Sharma & J. Bisht, ``Hermite--Hadamard-type inequalities for strongly $(alpha,m)$-convex functions via quantum calculus'', Journal of Applied Mathematics and Computing 70 (2024) 4971. https://doi.org/10.1007/s12190-024-02135-y.

[18] I. R. Silviya & K. Muthunagai, ``Differential and fuzzy differential sandwich theorems involving quantum calculus operators'', Journal of the Nigerian Society of Physical Sciences 6 (2024) 1832. https://doi.org/10.46481/jnsps.2024.1832.

[19] G. G. James, A. P. Ekong, A. U. Unyime, A. Akpanobong, J. A. Odey, D. O. Egete, S. Inyang, I. Ohaeri, C. M. Orazulume, E. Etuk & P. Okafor, ``Compiler-assisted code generation for quantum computing: leveraging the unique properties of quantum architectures'', Journal of the Nigerian Society of Physical Sciences 7 (2025) 2615. https://doi.org/10.46481/jnsps.2025.2615.

[20] N. Azzouz & B. Benaissa, ``Exploring Hermite--Hadamard-type inequalities via $psi$-conformable fractional integral operators'', Journal of Inequalities and Mathematical Analysis 1 (2025) 15. https://doi.org/10.63286/jima.2025.02.

[21] F. B. A-pai & H. Budak, ``Quantum Hermite--Hadamard-type inequalities for co-ordinated $(F,G)$-convex functions'', Journal of Computational and Applied Mathematics 475 (2026) 117030. https://doi.org/10.1016/j.cam.2025.117030.

[22] F. Wannalookkhee, K. Nonlaopon, S. K. Ntouyas, M. Z. Sar{i}kaya, H. Budak & M. A. Ali, ``Some new quantum Hermite--Hadamard inequalities for co-ordinated convex functions'', Mathematics 10 (2022) 1962. https://doi.org/10.3390/math10121962.

[23] F. Wannalookkhee, K. Nonlaopon, M. Z. Sar{i}kaya, H. Budak & M. A. Ali, ``On some new quantum trapezoid-type inequalities for $q$-differentiable coordinated convex functions'', Journal of Inequalities and Applications 2023 (2023) 5. https://doi.org/10.1186/s13660-023-02917-1.

[24] N. Alp, M. A. Ali, H. Budak & M. Z. Sar{i}kaya, ``New quantum bounds for $q$-Simpson-type inequalities for coordinated convex functions'', Filomat 36 (2022) 3919. https://www.jstor.org/stable/27385021.

[25] X.-X. You, M. A. Ali, G. Murtaza, S. Chasreechai, S. K. Ntouyas & T. Sitthiwirattham, ``Post-quantum Simpson's-type inequalities for coordinated convex functions'', AIMS Mathematics 7 (2022) 3097. https://doi.org/10.3934/math.2022172.

[26] H. Kalsoom, M. Vivas-Cortez & M. A. Latif, ``Trapezoidal-type inequalities for strongly convex and quasi-convex functions via post-quantum calculus'', Entropy 23 (2021) 1238. https://doi.org/10.3390/e23101238.

[27] H. Kalsoom & Z. A. Khan, ``New quantum boundaries for $q$-trapezoidal- and $q$-midpoint-type inequalities for $(eta_1,eta_2)$-convex functions'', Miskolc Mathematical Notes 26 (2025) 877. https://doi.org/10.18514/MMN.2025.4997.

[28] L. Zhang & R. Xu, ``Post-quantum Hermite--Hadamard-type inequalities for LR-$(h_1,h_2)$-convex interval-valued functions'', Arabian Journal of Mathematics 14 (2025) 365. https://doi.org/10.1007/s40065-025-00514-2.

[29] S. K. Mishra, R. Sharma & J. Bisht, ``Quantum analogue of trapezoid-type inequalities for $q$-differentiable coordinated strongly convex functions'', Journal of Applied Mathematics and Computing 71 (2025) 2473. https://doi.org/10.1007/s12190-024-02281-3.

[30] O. L. Mangasarian, Nonlinear programming, SIAM, Philadelphia, PA, USA, 1994. https://doi.org/10.1137/1.9781611971255.

[31] B. T. Polyak, ``Existence theorems and convergence of minimizing sequences for extremal problems with constraints'', Doklady Akademii Nauk SSSR 166 (1966) 287. https://www.mathnet.ru/php/archive.phtml?jrnid=dan&option_lang=eng&paperid=32017&wshow=paper.

[32] J. Hua, B.-Y. Xi & F. Qi, ``Some new inequalities of Simpson type for strongly $s$-convex functions'', Afrika Matematika 26 (2015) 741. https://doi.org/10.1007/s13370-014-0242-2.

[33] V. Kac & P. Cheung, Quantum calculus, Springer, New York, USA, 2002. https://doi.org/10.1007/978-1-4613-0071-7.

[34] F. H. Jackson, ``$q$-difference equations'', American Journal of Mathematics 32 (1910) 305. https://doi.org/10.2307/2370183.

[35] M. A. Latif, S. S. Dragomir & E. Momoniat, ``Some $q$-analogues of Hermite--Hadamard inequality for functions of two variables on finite rectangles in the plane'', Journal of King Saud University--Science 29 (2017) 263. https://doi.org/10.1016/j.jksus.2016.07.001.

[36] H. Budak, M. A. Ali & M. Tarhanac{i}, ``Some new quantum Hermite--Hadamard-like inequalities for coordinated convex functions'', Journal of Optimization Theory and Applications 186 (2020) 899. https://doi.org/10.1007/s10957-020-01726-6.