New Ostrowski Type Inequalities Using a 12-Step Peano Kernel: Introduction and Applications to AI Reliability

Source 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 L 2 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 L 2 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.