Hermite-Hadamard-Fejér Inequalities for Conformable Fractional Integrals via Preinvex Functions

Yousaf Khurshid,Zareen Abdulhameed Khan,Muhammad Adil Khan,Yu-Ming Chu

doi:10.1155/2019/3146210

Abstract

In this paper, we present a Hermite-Hadamard-Fejér inequality for conformable fractional integrals by using symmetric preinvex functions. We also establish an identity associated with the right hand side of Hermite-Hadamard inequality for preinvex functions; then by using this identity and preinvexity of functions and some well-known inequalities, we find several new Hermite-Hadamard type inequalities for conformal fractional integrals.

Highlights

Let I ∈ R be an interval and h : I 󳨀→ R be a convex function defined on I such that κ1, κ2 ∈ I with κ1 < κ2
We present a Hermite-Hadamard-Fejer inequality for conformable fractional integrals by using symmetric preinvex functions
We establish an identity associated with the right hand side of Hermite-Hadamard inequality for preinvex functions; by using this identity and preinvexity of functions and some well-known inequalities, we find several new Hermite-Hadamard type inequalities for conformal fractional integrals

Summary

Introduction

Let I ∈ R be an interval and h : I 󳨀→ R be a convex function defined on I such that κ1, κ2 ∈ I with κ1 < κ2. In 1906, Fejer [16] established the following weighted generalization of the Hermite-Hadamard inequality for symmetric functions: h The conformable fractional derivative of order 0 < α ≤ 1 at s > 0 for the function h : [0, ∞) 󳨀→ R is defined by h (s + εs1−α) − h (s). We present some basic results related to conformable fractional derivative in the following theorem. We first establish the Hermite-HadamardFejer inequality for conformable fractional integrals by using symmetric preinvex functions; we present HH inequalities as their special cases (see Corollary 9). We give an identity associated with the right side of HH inequality for preinvex functions using the conformable fractional integrals; we establish HH inequalities for preinvex functions by use of Holder inequality, power mean inequality, and preinvexity of functions

Hermite-Hadamard-Fejér Inequalities for Conformable Fractional Integrals

HH Type Inequalities for Conformable Fractional Integrals