Wirtinger-type inequalities for Caputo fractional derivatives via Taylor’s formula

Abstract

In this study, we firstly derive a Wirtinger-type result, which gives the connection in between the integral of square of a function and the integral of square of its Caputo fractional derivatives with the help of left-sided and right-sided fractional Taylor’s Formulas. Afterward, we provide a more general inequality involving Caputo fractional derivatives for $L_{r}$ norm with $r>1$ via Hölder’s inequality. Also, similar inequalities for Riemann–Liouville fractional derivatives are presented by means of a relation between Caputo fractional derivatives and Riemann–Liouville fractional derivatives.