Error bounds of multiplicative Boole’s type inequalities for twice differentiable functions with applications to numerical integration

Abstract

This paper presents significant advancements in multiplicative calculus by establishing refined inequalities for Boole’s formula applied to twice-differentiable functions. It is rigorously demonstrated that the derived inequalities provide improved error bounds for integral approximations,
thereby enhancing the utility of Boole’s formula in numerical methods. By adapting Boole’s formula to the framework of multiplicative calculus, new insights are revealed, highlighting relationships and properties specifically tailored to this modern calculus. This adaptation improves the precision of integral approximations and broadens the scope of applications where such methods can be effectively employed. One of the key findings in this study is that multiplicative calculus provides significantly better absolute error bounds compared to classical calculus, particularly for higher-degree polynomials. Moreover, applications to quadrature formulas and special means for real numbers within the framework of multiplicative calculus demonstrate the
practical utility of the newly derived inequalities. To illustrate the main findings, numerical examples with graphical representations are provided.