A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions

Sanja Kovač

doi:10.3390/axioms11010016

Abstract

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained.

Highlights

IntroductionThe main objects of our interest are integral inequalities involving weight functions of Hermite–Hadamard–Fejér type for various classes of convex functions
Let us start by bringing out the well-known definition of convex functions: Definition 1
In [1], Toader established the class of m-convex functions as the following: Chesneau

Summary

Introduction

The main objects of our interest are integral inequalities involving weight functions of Hermite–Hadamard–Fejér type for various classes of convex functions. In [1], Toader established the class of m-convex functions as the following: Chesneau. Fejér [4] has obtained the weighted version of famous Hermite–Hadamard inequality: Theorem 1. Many researchers have been interested in improving and refining Hermite–Hadamard inequalities for various types of convex functions (see for instance [5,6,7,8,9]). In [10], authors established a new integral inequalitites of Hermite–Hadamard type for (α, m)-function. Fejér type inequalities for n-times differentiable functions which are m-convex and (α, m)convex. Some special cases for different choices of weight functions are given

Main Results

Applications to Special Means

Conclusions