Some Fractional Hermite–Hadamard Type Inequalities for Interval-Valued Functions

Fangfang Shi,Dafang Zhao,Wei Liu,Guoju Ye

doi:10.3390/math8040534

Fangfang Shi, Dafang Zhao + Show 2 more

Open Access

https://doi.org/10.3390/math8040534

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Journal: Mathematics	Publication Date: Apr 4, 2020
Citations: 8	License type: CC BY 4.0

Affiliation: Hohai University, Hubei Normal University

Abstract

In this paper, firstly we prove the relationship between interval h-convex functions and interval harmonically h-convex functions. Secondly, several new Hermite–Hadamard type inequalities for interval h-convex functions via interval Riemann–Liouville type fractional integrals are established. Finally, we obtain some new fractional Hadamard–Hermite type inequalities for interval harmonically h-convex functions by using the above relationship. Also we discuss the importance of our results and some special cases. Our results extend and improve some previously known results.

Highlights

Hermite–Hadamard inequality was firstly discovered by Hermite and Hadamard for convex functions are considerable significant in the literature
Interval analysis was initially developed as an attempt to deal with interval uncertainty that appears in computer graphics [9], automatic error analysis [10], and many others
Hermite–Hadamard type inequalities involving interval Riemann–Liouville type fractional integral have been obtained for different classes of interval-valued functions (IVFs), see for interval convex functions [17], for interval harmonically convex functions [18] and the references therein

Summary

Introduction

Hermite–Hadamard inequality was firstly discovered by Hermite and Hadamard for convex functions are considerable significant in the literature. Many papers relating to fractional integral inequalities have been obtained for different classes of functions, see [6,7,8]. Hermite–Hadamard type inequalities involving interval Riemann–Liouville type fractional integral have been obtained for different classes of IVFs, see for interval convex functions [17], for interval harmonically convex functions [18] and the references therein. Motivated by the ongoing research, We proved the relationship between interval h-convex functions and interval harmonically h-convex functions, we establish some new Hermite–Hadamard type inequalities for interval h-convex functions and interval harmonically h-convex functions via interval Riemann–Liouville type fractional integrals. Results obtained in this paper may be extended for other classes of convex functions. Mathematics 2020, 8, 534 including interval (h1 , h2 )-convex functions and interval Log-h-convex functions and used as a tool to investigate the research of optimization and probability, among others

Preliminaries and Result

Fractional Hermite–Hadamard Type Inequalities of Interval h-Convex Functions

Conclusions