Weighted Midpoint Hermite-Hadamard-Fejér Type Inequalities in Fractional Calculus for Harmonically Convex Functions

Humaira Kalsoom,Muhammad Amer Latif,Miguel Vivas-Cortez,Hijaz Ahmad

doi:10.3390/fractalfract5040252

Abstract

In this paper, we establish a new version of Hermite-Hadamard-Fejér type inequality for harmonically convex functions in the form of weighted fractional integral. Secondly, an integral identity and some weighted midpoint fractional Hermite-Hadamard-Fejér type integral inequalities for harmonically convex functions by involving a positive weighted symmetric functions have been obtained. As shown, all of the resulting inequalities generalize several well-known inequalities, including classical and Riemann–Liouville fractional integral inequalities.

Highlights

The theory of convex functions is an essential tool in various fields of pure and applied sciences
The Hermite-Hadamard integral inequality is a well-known inequality in the subject of convex functional analysis
The extraordinary inequality states that if F : I → R is a convex mapping on the interval I of real numbers and θ1, θ2 ∈ I with θ1 < θ2

Summary

Introduction

The theory of convex functions is an essential tool in various fields of pure and applied sciences. İşcan et al published Hermite-inequality Hadamard’s in fractional integral type for harmonically convex functions In [22], as follows: Theorem 2. Hermite-Hadamard inequalities for harmonically convex functions were introduced in fractional integral form in [25] as follows: Theorem 3. In [17] Chan et al stated the Hermite-HadamardFejér inequality as follows: Theorem 4. In [26], İşcan et al proved Hermite-Hadamard-Fejér type inequalities for harmonically convex functions through fractional integrals: Theorem 5. We shall prove our key results, which will include new midpoint fractional Hermite-Hadamard-Fejér type integral inequalities as well as some related results.

Auxiliary Results

Main Results

Conclusions