Generalized Alomari functionals

TL;DR

This paper introduces a generalized Alomari functional that unifies various classical integral inequalities and quadrature rules, providing new bounds and applications for numerical integration.

Contribution

It extends existing inequalities to a generalized functional framework, encompassing trapezoidal, midpoint, Simpson, and Newton-Simpson rules as special cases.

Findings

Derived inequalities for the generalized Alomari functional using the n-th order modulus.

Established bounds for classical quadrature rules within the generalized framework.

Applied the inequalities to improve error estimates in numerical integration.

Abstract

We consider a generalized form of certain integral inequalities given by Guessab, Schmeisser and Alomari. The trapezoidal, mid point, Simpson, Newton-Simpson rules are obtained as special cases. Also, inequalities for the generalized Alomari functional in terms of the $n$-th order modulus, $n=\overline{1,4}$, are given and applied to some known quadrature rules.