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

Abstract

It is well known that every convex function f : I → R (where I ⊂ R is an interval) admits an affine support at every interior point of I (i.e. for any x 0 ∈ Int I there exists an affine function a : I → R such that a ( x 0 ) = f ( x 0 ) and a ⩽ f on I). 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.