The aim of this paper is to present new more general Hardy-type inequalities for different kinds of fractional integrals and fractional derivatives.

In this paper, we extend Hardy’s type inequalities to convex functions of higher order. Upper bounds for the generalized Hardy’s inequality are given with some applications.

In this article, we state, prove and discuss new Steffensen type fractional inequalities involving fractional derivatives of Canavati, Riemann–Liouville and Caputo types. Finally, we define functionals as a difference between the right- and left-hand sides of the new fractional Steffensen type inequalities and obtain related analogous of the Lagrange and the Cauchy mean value theorems. Furthermore, exponential convexity is proven.