Ostrowski-type integral inequalities provide computable bounds for the error between function values, weighted sums, and integral means, but classical estimates based on a single global derivative bound can be conservative. This study introduces a symmetric 12-step Peano kernel and develops new Ostrowski-type inequalities for functions satisfying three regularity conditions. A fundamental integral identity is established by applying integration by parts over the twelve kernel subintervals. The resulting remainders are bounded using the Gruss, Cauchy-Schwarz, and Diaz-Metcalf inequalities in the relevant function spaces. The sharpest L2 estimate is then applied to cumulative distribution functions on bounded intervals to construct a Certified Expectation Estimator (CEE). The estimator combines a fixed weighted set of CDF evaluations with the L2 norm of the probability density function to approximate the expectation and produce a deterministic, non-asymptotic a priori error bound. Numerical examples involving uniform, Beta(2,2), and truncated normal distributions illustrate that the bound adapts to the density norm and follows the predicted dependence on the interval length. The proposed framework links classical integral inequality theory with certified computation and provides a reproducible method for expectation estimation in applications such as Bayesian inference, reinforcement learning, and neural network verification. The results demonstrate that the symmetric 12-step construction offers a practical balance between analytical tractability, computational cost, and rigorous reliability guarantees.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 12, Issue 4) |
| DOI | 10.11648/j.ijtam.20261204.12 |
| Page(s) | 80-89 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Ostrowski Inequality, Peano Kernel, 12-step Kernel, Certified Expectation Estimator, Cumulative Distribution Function, Numerical Integration, Bayesian Inference, Trustworthy Artificial Intelligence
| [1] | A. Ostrowski, Uber die Absolutabweichung einer differentiierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10 (1938), 226-227. |
| [2] | S. S. Dragomir and T. M. Rassias (eds.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002. |
| [3] | M. Muawwaz, M. Maaz, A. Qayyum, M. Ahmad, M. D. Faiz, and A. Mehboob, Innovative Ostrowski's type inequalities based on linear kernel and applications, Palestine J. Math., 14(1) (2025), 802-812. |
| [4] | M. Maaz, M. Muawwaz, U. Ali, M. D. Faiz, E. A. Rahman, and A. Qayyum, New extension in Ostrowski's type inequalities by using 13-step linear kernel, Adv. Anal. Appl. Math., 1(1) (2024), 55-67. |
| [5] | A. Munir, M. Vivas-Corte, A. Qayyum, H. Budak, I. Faiz, and S. S. Supadi, Some new fractional corrected Euler-Maclaurin type inequalities for functions whose second derivatives are s-convex, Math. Comput. Model. Dyn. Syst., 30(1) (2024). |
| [6] | A. Munir, A. Qayyum, L. Rathour, G. Atta, S. S. Supadi, and U. Ali, A study on Milne-type inequalities for a specific fractional integral operator with applications, Korean J. Math., 32(2) (2024), 297-314. |
| [7] | M. W. Alomari, A companion of Ostrowski's inequality with applications, Transylv. J. Math. Mech., 3 (2011), 9-14. |
| [8] | M. W. Alomari, A companion of Ostrowski's inequality for mappings whose first derivatives are bounded and applications in numerical integration, Kragujevac J. Math., 36 (2012), 77-82. |
| [9] | H. Budak, M. Z. Sarikaya, and A. Qayyum, Improvement in companion of Ostrowski type inequalities for mappings whose first derivatives are of bounded variation and applications, Filomat, 31 (2017), 5305-5314. |
| [10] | A. Qayyum, I. Faye, and M. Shoaib, Improvement of Ostrowski integral type inequalities with application, Filomat, 30(6) (2016), 1441-1456. |
| [11] | W. Liu, Y. Zhu, and J. Park, Some companions of perturbed Ostrowski-type inequalities based on the quadratic kernel function with three sections and applications, J. Inequal. Appl., 2013:226 (2013). |
| [12] | M. Karim, A. Fahmi, S. Qaisar, Z. Ullah, and A. Qayyum, New developments in fractional integral inequalities via convexity with applications, AIMS Math., 8(7) (2023), 15950-15968. |
| [13] | A. Qayyum, M. Shoaib, and M. A. Latif, A generalized inequality of Ostrowski type for twice differentiable bounded mappings and applications, Appl. Math. Sci., 8(38) (2014), 1889-1901. |
| [14] | A. Munir, A. Qayyum, H. Budak, S. Qaisar, U. Ali, and S. S. Supadi, A fractional version of corrected dual-Simpson's type inequality via s-convex function with applications, Malaysian J. Math. Sci., 19(1) (2025), 17-33. |
| [15] | G. Gruss, On the maximum of the absolute value of the difference between the integral mean of a product and the product of the integral means, Math. Z., 39 (1935), 215? 226. |
| [16] | D. S. Mitrinovi'{c}, J. E. Pev{c}ari'{c}, and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993. |
| [17] | S. I. Butt, M. Nadeem, and M. U. Awan, Parameterized fractal-fractional analysis of Ostrowski- and Simpson-type inequalities, Fractal Fract., 9(1) (2025), 15. |
APA Style
Faiz, M. D., Talhat, R., Amjad, J., Qayyum, A., Shabir, G. (2026). New Ostrowski Type Inequalities Using a 12-Step Peano Kernel: Introduction and Applications to AI Reliability. International Journal of Theoretical and Applied Mathematics, 12(4), 80-89. https://doi.org/10.11648/j.ijtam.20261204.12
ACS Style
Faiz, M. D.; Talhat, R.; Amjad, J.; Qayyum, A.; Shabir, G. New Ostrowski Type Inequalities Using a 12-Step Peano Kernel: Introduction and Applications to AI Reliability. Int. J. Theor. Appl. Math. 2026, 12(4), 80-89. doi: 10.11648/j.ijtam.20261204.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ijtam.20261204.12,
author = {Muhammad Danial Faiz and Rafia Talhat and Junaid Amjad and Ather Qayyum and Ghulam Shabir},
title = {New Ostrowski Type Inequalities Using a 12-Step Peano Kernel: Introduction and Applications to AI Reliability},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {12},
number = {4},
pages = {80-89},
doi = {10.11648/j.ijtam.20261204.12},
url = {https://doi.org/10.11648/j.ijtam.20261204.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20261204.12},
abstract = {Ostrowski-type integral inequalities provide computable bounds for the error between function values, weighted sums, and integral means, but classical estimates based on a single global derivative bound can be conservative. This study introduces a symmetric 12-step Peano kernel and develops new Ostrowski-type inequalities for functions satisfying three regularity conditions. A fundamental integral identity is established by applying integration by parts over the twelve kernel subintervals. The resulting remainders are bounded using the Gruss, Cauchy-Schwarz, and Diaz-Metcalf inequalities in the relevant function spaces. The sharpest L2 estimate is then applied to cumulative distribution functions on bounded intervals to construct a Certified Expectation Estimator (CEE). The estimator combines a fixed weighted set of CDF evaluations with the L2 norm of the probability density function to approximate the expectation and produce a deterministic, non-asymptotic a priori error bound. Numerical examples involving uniform, Beta(2,2), and truncated normal distributions illustrate that the bound adapts to the density norm and follows the predicted dependence on the interval length. The proposed framework links classical integral inequality theory with certified computation and provides a reproducible method for expectation estimation in applications such as Bayesian inference, reinforcement learning, and neural network verification. The results demonstrate that the symmetric 12-step construction offers a practical balance between analytical tractability, computational cost, and rigorous reliability guarantees.},
year = {2026}
}
TY - JOUR T1 - New Ostrowski Type Inequalities Using a 12-Step Peano Kernel: Introduction and Applications to AI Reliability AU - Muhammad Danial Faiz AU - Rafia Talhat AU - Junaid Amjad AU - Ather Qayyum AU - Ghulam Shabir Y1 - 2026/08/25 PY - 2026 N1 - https://doi.org/10.11648/j.ijtam.20261204.12 DO - 10.11648/j.ijtam.20261204.12 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 80 EP - 89 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20261204.12 AB - Ostrowski-type integral inequalities provide computable bounds for the error between function values, weighted sums, and integral means, but classical estimates based on a single global derivative bound can be conservative. This study introduces a symmetric 12-step Peano kernel and develops new Ostrowski-type inequalities for functions satisfying three regularity conditions. A fundamental integral identity is established by applying integration by parts over the twelve kernel subintervals. The resulting remainders are bounded using the Gruss, Cauchy-Schwarz, and Diaz-Metcalf inequalities in the relevant function spaces. The sharpest L2 estimate is then applied to cumulative distribution functions on bounded intervals to construct a Certified Expectation Estimator (CEE). The estimator combines a fixed weighted set of CDF evaluations with the L2 norm of the probability density function to approximate the expectation and produce a deterministic, non-asymptotic a priori error bound. Numerical examples involving uniform, Beta(2,2), and truncated normal distributions illustrate that the bound adapts to the density norm and follows the predicted dependence on the interval length. The proposed framework links classical integral inequality theory with certified computation and provides a reproducible method for expectation estimation in applications such as Bayesian inference, reinforcement learning, and neural network verification. The results demonstrate that the symmetric 12-step construction offers a practical balance between analytical tractability, computational cost, and rigorous reliability guarantees. VL - 12 IS - 4 ER -
Department of Mathematics, University of Southern Punjab, Multan, Pakistan
Department of Mathematics, University of Southern Punjab, Multan, Pakistan
Department of Mathematics, University of Southern Punjab, Multan, Pakistan
Department of Mathematics, University of Southern Punjab, Multan, Pakistan
Department of Mathematics, University of Southern Punjab, Multan, Pakistan
Information