Support-type properties of convex functions of higher order and Hadamard-type inequalities

TL;DR

This paper extends the concept of support properties from convex functions to higher-order convex functions, developing a new method to derive Hadamard-type inequalities and error bounds for quadrature rules.

Contribution

It introduces the attaching method for higher-order convex functions, generalizes support theorems, and derives new inequalities and error bounds for numerical integration.

Findings

Higher-order convex functions are supported by polynomials of degree no greater than their order.

The attaching method leads to a general support theorem for higher-order convex functions.

New Hadamard-type inequalities and error bounds for quadrature rules are established.

Abstract

It is well-known that every convex function admits an affine support at every interior point of a domain. Convex functions of higher order (precisely of an odd order) have a similar property: they are supported by the polynomials of degree no greater than the order of convexity. In this paper the attaching method is developed. It is applied to obtain the general result Theorem 2, from which the mentioned above support theorem and some related properties of convex functions of higher (both odd and even) order are derived. They are applied to obtain some known and new Hadamard-type inequalities between the quadrature operators and the integral approximated by them. It is also shown that the error bounds of quadrature rules follow by inequalities of this kind.