Family of inequalities for Newton-Cotes formulae with their error bounds

Abstract

In numerical analysis, quadrature formulae are essential for approximating definite integrals. This paper introduces a new family of Newton-Cotes-type inequalities, derived from a parameterized identity, and establishes their associated error bounds for several function classes (Convex, Bounded, and Lipschitzian). A key advantage of these inequalities is their generality: they provide a unified framework to derive the error bounds of classical quadrature formulae such as the Midpoint, Simpson’s $1/3$, Simpson’s $3/8$, Maclaurin’s, and Weddle’s approximations. We extend traditional formulae and employ the power mean inequality and Hölder’s integral inequality to obtain more general and sharp error estimates. The practical utility of these inequalities is demonstrated through solutions to mathematical problems and a precise numerical assessment comparing their efficiency in realistic examples. Applications to quadrature formulae and numerical examples further illustrate the effectiveness of the proposed inequalities, contributing to the advancement of numerical methods and broadening the applicability of Newton-Cotes-type inequalities within mathematical sciences.