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.

Support-type properties of generalized convex functions

“…Section 3 contains the first main result of the paper, which is Theorem 3.1 of support-type. Its special case concerning higher-order convexity was recently proved in [24,Theorem 2]. In Section 4 we present some applications of Theorem 3.1.…”

“…They can be used to prove the HermiteHadamard-type inequalities between the quadrature operators and the integral of the function, which is convex in a desired sense. Using the support approach, for higher-order convexity it was done by the present author in [24], for (u 0 , u 1 )-convexity by Bessenyei and Páles [6] and for convexity with respect to Beckenbach families by Bessenyei [4]. There is also a series of papers, where the results of this kind were proved using another approach: [2] by Bessenyei and [5,7,8] by Bessenyei and Páles.…”

“…The method used in the proof given in [24] is based on boundedness of divided differences and the Newton's form of the interpolating polynomial. The constructed polynomial is in fact the same as the polynomial obtained in the proof of Theorem 3.1 by osculatory interpolation (in the special case of the polynomial ECT-system).…”

“…For instance they can be used to prove Hermite-Hadamard-type inequalities between quadrature operators (the well known Gauss-Legendre, Lobatto and Radau quadratures in the case of higher-order convexity) and the integral of the U -convex function. For details see [24] (higherorder convexity) and [4] (the general U -convexity).…”

Approximation orders and shape preserving properties of the multiquadric trigonometric B-spline quasi-interpolant

“…Furthermore, according to Wasowicz [45], the shape properties are equivalent to the nonnegativity of the corresponding divided differences that are discretizations of the derivatives. For example, the monotonicity and convexity are synonymous with the nonnegativity of the first-order and second-order divided differences, respectively.…”