Trapezium-Type Inequalities for Raina’s Fractional Integrals Operator Using Generalized Convex Functions

Miguel Vivas-Cortez,Artion Kashuri,Jorge E Hernández Hernández

doi:10.3390/sym12061034

Miguel Vivas-Cortez, Artion Kashuri + Show 1 more

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https://doi.org/10.3390/sym12061034

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Abstract

The authors have reviewed a wide production of scientific articles dealing with the evolution of the concept of convexity and its various applications, and based on this they have detected the relationship that can be established between trapezoidal inequalities, generalized convex functions, and special functions, in particular with the so-called Raina function, which generalizes other better known ones such as the hypergeometric function and the Mittag–Leffler function. The authors approach this situation by studying the Hermite–Hadamard inequality, establishing a useful identity using Raina’s fractional integral operator in the setting of ϕ -convex functions, obtaining some integral inequalities connected with the right-hand side of Hermite–Hadamard-type inequalities for Raina’s fractional integrals. Various special cases have been identified.

Highlights

In recent decades, the concept of convexity has had a more general evolution due to its wide application in various fields of science, as demonstrated in the following works [1,2,3,4]
The main objective of this paper is to establish Hermite–Hadamard’s inequalities for Raina’s fractional integral operator using a similar method in [35] via generalized φ-convex functions and we will investigate some integral inequalities connected with the right-hand side of the Hermite–Hadamard type inequalities for Raina’s fractional integrals
In the development of this work, a special definition of generalized φ-convex functions has been introduced when considering the class of functions defined by R.K

Summary

Introduction

The concept of convexity has had a more general evolution due to its wide application in various fields of science, as demonstrated in the following works [1,2,3,4]. Among the types of generalized convexity and its applications are some such as h-convexity, MT-convexity, η −convexity,. The use of special functions, in addition to those involving fractional integral operators, have been related to this topic [6,10,11,12]. The following inequality, named Hermite–Hadamard inequality, is one of the most famous inequalities in the literature for convex functions. Let f : I ⊆ R −→ R be a convex function and p, q ∈ I with p < q.

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