Error Bounds of Boole's Formula type Inequalities forGeneralized Coordinated Convex Functions with ComputationalAnalysis and Applications

Abstract

This study aims to enhance the error bounds for Boole's formula-type inequalities applied to generalized coordinate convex functions. A solid mathematical foundation is established to provide improved error estimates for two-variable integrals that surpass those derived from traditional coordinate convex functions. The motivation is twofold: to extend results from convex functions to the broader class of s-convex functions and to o er optimal approximations that enhance numerical method accuracy. Moreover, Boole's formula o ers better absolute error bounds for higher-degree polynomials, especially in cases where other methods, such as Simpson's rule, fail to provide the required precision. This is due to Boole's formula's ability to approximate polynomials up to degree  ve exactly, whereas Simpson's rule is limited to polynomials of degree
three. Through computational analysis, key results are presented, demonstrating that the improved error bounds provide greater reliability for numerical integration applications. This research broadens the theoretical framework of mathematical analysis and optimization while
providing practical methodologies to tackle real-world challenges with accuracy and e ectiveness. Future extensions of this work may include applications to q-calculus, symmetrized q-calculus, fractional calculus, and multidimensional spaces, providing deeper insights into the behavior of
Boole's formula and its error bounds.