On some extremalities in the approximate integration

TL;DR

This paper establishes new inequalities for quadrature operators based solely on higher order convexity, refining Hadamard-type inequalities and providing weaker error bounds without differentiability assumptions.

Contribution

It introduces extremal inequalities for quadrature operators relying only on higher order convexity, avoiding traditional differentiability conditions.

Findings

Refined inequalities of Hadamard type for convex functions

Error bounds for quadrature operators under weaker assumptions

No differentiability assumptions needed for the inequalities

Abstract

Some extremalities for quadrature operators are proved for convex functions of higher order. Such results are known in the numerical analysis, however they are often proved under suitable differentiability assumptions. In our considerations we do not use any other assumptions apart from higher order convexity itself. The obtained inequalities refine the inequalities of Hadamard type. They are applied to give error bounds of quadrature operators under the assumptions weaker from the commonly used.