On some inequalities involving liouville-caputo fractional derivatives and applications to special means of real numbers

Abstract

We are concerned with the class of functions f ∈ C1([a, b];ℝ), a, b ∈ ℝ, a < b, such that |cD aα f j is convex or| cDbα f|is convex, where 0 < α < 1, cD aα f is the left-side Liouville-Caputo fractional derivative of order α of f and cD bα f is the right-side Liouville-Caputo fractional derivative of order α of f . Some extensions of Dragomir-Agarwal inequality to this class of functions are obtained. A parallel development is made for the class of functions f ∈ C1([a, b];ℝ) such that |cDaα f| is concave or cD bα f | is concave. Next, an application to special means of real numbers is provided.