Boole-Type Inequalities and Improved Error Estimates for Quadrature Formulae via Twice Differentiable Convex Functions

Abstract

New inequalities presented by investigating quadrature formulae with their error bounds for twice differentiable convex functions. Being inspired by the central role of quadrature formulae for estimating definite integrals, this work proposes new Boole's identity for twice differentiable functions. By using the newly established identity, Boole's type inequalities for second-times differentiable convex functions are proved. By applying the power mean and Holder's inequalities, the present research improved the error estimates for Boole's formula. The theoretical advancements are validated through numerical examples involving exponential and polynomial functions, demonstrating the efficacy of the proposed inequalities. Additionally, applications to the q-digamma function and modified Bessel functions underscore the versatility of the results. This work not only refines existing quadrature error analyses but also expands the applicability of convex function theory in numerical integration, contributing to both inequality theory and computational mathematics.