Results on integral inequalities for a generalized fractional integral operator unifying two existing fractional integral operators

Abstract

The main aim of this article is to design a novel framework to study a generalized fractional integral operator that unifies two existing fractional integral operators. To ensure the suitable selection of the operator and with the discussion of special cases, it is shown that our considered fractional integral generalizes the well-known Atangana–Baleanu fractional integral (AB-fractional integral) and the ABK-fractional integral. Conditions are stated for the generalized AB-fractional integral operator (GAB-fractional integral operator) to be bounded in the space X_c^p (γ₁,γ₂). We also provide a fractional product-integration formula for this operator. Furthermore, we generalize the reverse Minkowski’s inequality and the reverse Hölder-type inequality by utilizing the GAB-fractional integral operator. Additionally, some other types of integral inequalities are established, and several special cases are noted. The concepts in this article may influence further research in fractional calculus.