Hermite-Hadamard and Hermite-Hadamard-Fej\'er type Inequalities for Generalized Fractional Integrals

TL;DR

This paper develops generalized Hermite-Hadamard and Hermite-Hadamard-Fejér inequalities for fractional integrals that unify Riemann-Liouville and Hadamard types, providing a broader framework with limiting cases.

Contribution

It introduces a unified form of fractional integrals encompassing Riemann-Liouville and Hadamard integrals, deriving new inequalities within this generalized setting.

Findings

Derived Hermite-Hadamard inequalities for the generalized fractional integrals.

Showed that classical Riemann-Liouville and Hadamard inequalities are special cases.

Established limits connecting the generalized form to known fractional integrals.

Abstract

In this paper we obtain the Hermite-Hadamard and Hermite-Hadamard-Fej\'er type inequalities for fractional integrals which generalize the two familiar fractional integrals namely, the Riemann-Liouville and the Hadamard fractional integrals into a single form. We prove that, in most cases, we obtain the Riemann--Liouville and the Hadamard equivalence just by taking limits when a parameter $\rho \rightarrow 1$ and $\rho \rightarrow 0^+$, respectively.