Hermite–Hadamard–Fejér-Type Inequalities and Weighted Three-Point Quadrature Formulae

Mihaela Ribičić Penava

doi:10.3390/math9151720

Abstract

The goal of this paper is to derive Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and a general three-point integral formula involving harmonic sequences of polynomials and w-harmonic sequences of functions. In special cases, Hermite–Hadamard–Fejér-type estimates are derived for various classical quadrature formulae such as the Gauss–Legendre three-point quadrature formula and the Gauss–Chebyshev three-point quadrature formula of the first and of the second kind.

Highlights

The importance and significance of our paper are reflected in the way in which we prove new Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and the general weighted three-point quadrature formula by using inequality (1), and a weighted version of the integral identity expressed by w-harmonic sequences of functions
Let us introduce the notations and terminology used in relation to w-harmonic sequences of functions
We establish Hermite–Hadamard–Fejér-type inequalities for the weighted three-point formula using a weighted version of the integral identity expressed by wharmonic sequences of functions that are given in Theorem 2 and the method that originated in [15]

Summary

Introduction

The Hermite–Hadamard inequalities and their weighted versions, the so-called HermiteHadamard-Fejér inequalities, are the most well-known inequalities related to the integral mean of a convex function (see [1] (p. 138)). The importance and significance of our paper are reflected in the way in which we prove new Hermite–Hadamard–Fejér-type inequalities for higher-order convex functions and the general weighted three-point quadrature formula by using inequality (1), and a weighted version of the integral identity expressed by w-harmonic sequences of functions. For this purpose, let us introduce the notations and terminology used in relation to w-harmonic sequences of functions (see [14]). We assume that all considered integrals exist and that they are finite

Hermite–Hadamard–Fejér-Type Inequalities for Three-Point Quadrature Formulae

Special Cases

Gauss–Legendre Three-Point Quadrature Formula h

Gauss–Chebyshev Three-Point Quadrature Formula of the First Kind

Gauss–Chebyshev Three-Point Quadrature Formula of the Second Kind