Conformable Fractional Integrals Versions of Hermite-Hadamard Inequalities and Their Generalizations

Muhammad Adil Khan,Artion Kashuri,Yu-Ming Chu,Gohar Ali,Rozana Liko

doi:10.1155/2018/6928130

Abstract

We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity and by using this identity we obtain generalized Hermite-Hadamard type inequalities.

Highlights

The class of convex functions is well known in the literature and is usually defined in the following way: let I be an interval in R; a function f : I → R is said to be convex on I if the inequality f (λx + (1 − λ) y) ≤ λf (x) + (1 − λ) f (y) (1)holds for all x, y ∈ I and λ ∈ [0, 1]
We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions
In the paper [1], Hudzik and Maligranda considered a generalization of convex function, which is known as s-convex function in the second sense

Summary

Introduction

The class of convex functions is well known in the literature and is usually defined in the following way: let I be an interval in R; a function f : I → R is said to be convex on I if the inequality f (λx + (1 − λ) y) ≤ λf (x) + (1 − λ) f (y) (1). Both inequalities hold in reverse direction if the function f is concave on the interval I This remarkable result was given in ([3], 1893) and is well known in the literature as Hermite-Hadamard inequality. Anderson [21] investigated the following conformable integral version of Hermite-Hadamard inequality. We prove new Hermite-Hadamard inequalities for conformable fractional integrals by using convex function, s-convex, and coordinate convex functions. We prove new Montgomery identity for conformable fractional integral By using this identity, we obtain Hermite-Hadamard type inequalities.

Hermite-Hadamard Inequalities

Generalization of Hermite-Hadamard Type Inequalities