Abstract
Fractional operators are one of the frequently used tools to obtain new generalizations of clasical inequalities in recent years and many new fractional operators are defined in the literature. This development in the field of fractional analysis has led to a new orientation in various branches of mathematics and in many of the applied sciences. Thanks to this orientation, it has brought a whole new dimension to the field of inequality theory as well as many other disciplines. In this study, a new lemma has been proved for the fractional integral operator defined by Atangana and Baleanu. Later with the help of this lemma and known inequalities such as Young, Jensen, Hölder, new generalizations of Hermite-Hadamard inequality are obtained. Many reduced corollaries about the main findings are presented for classical integrals.
Highlights
First of all, let us recall the concept of convex function which is the basic concept of convex analysis.Definition 1
We will start with a new integral identity that will be used the proofs of our main findings as following: Lemma 1. κ : [μ, ν] → R be differentiable function on (μ, ν) with μ < ν
In the introduction part, a historical background in the field of inequality theory and fractional analysis is presented, and in the main results part, a new integral equation is produced based on fractional integral operators
Summary
Let us recall the concept of convex function which is the basic concept of convex analysis.Definition 1. Several new results have been proved related different kinds of convex functions and associated integral inequalities. In [6], Sarikaya et al gave a different perspective to the inequality (2) by using the Riemann-Liouville fractional integral operators as follows: Theorem 2 ([6]).
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