Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities

We prove that if $f:I\subset \mathbb{R}\to \mathbb{R}$ is of bounded variation, then the uncentered maximal function $Mf$ is absolutely continuous, and its derivative satisfies the sharp inequality $\Vert DMf\Vert _{L^1(I)}\le \vert Df\vert(I)$. This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.