Some New Fractional Estimates of Inequalities for LR-p-Convex Interval-Valued Functions by Means of Pseudo Order Relation

Muhammad Bilal Khan,Dumitru Baleanu,Muhammad Aslam Noor,Pshtiwan Othman Mohammed,Juan Luis García Guirao

doi:10.3390/axioms10030175

Abstract

It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis, both the inclusion relation (⊆) and pseudo order relation (≤p) are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-p-convex interval-valued functions (LR-p-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-p-convex-IVFs and Hermite-Hadamard type inequalities (HH-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-p-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area.

Highlights

Hermite [1] and Hadamard [2] derived the familiar inequality known as HermiteHadamard inequality (HH inequality)
Inspired by the ongoing research work, we generalize the class of p-convex function known as LR-p-convex-IVF, and establish the relationship between HH-type inequalities and LR-p-convex-IVF via Katugampola fractional integral
The interval left and right Katugampola fractional integrals of f ∈ =L[u,ν] with order are defined by p,α

Summary

Introduction

Hermite [1] and Hadamard [2] derived the familiar inequality known as HermiteHadamard inequality (HH inequality). Fejér [16] considered the major generalization of HH-inequality which is known as HH-Fejér inequality It can be expressed as follows: Let f : [u, ν] → R be a convex function on an interval [u, ν] with u ≤ ν , and let. Due to the vast applications of convexity and fractional HH-inequality in mathematical analysis and optimization, many authors have discussed the applications, refinements, generalizations, and extensions, see [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56] and the references therein. Inspired by the ongoing research work, we generalize the class of p-convex function known as LR-p-convex-IVF, and establish the relationship between HH-type inequalities and LR-p-convex-IVF via Katugampola fractional integral

Preliminaries

LR-p-Convex Interval-Valued Functions

Conclusions