Sharp derivative inequalities for maximal functions

We determine the optimal constant in the $L^p$-derivative inequality for the uncentered Hardy–Littlewood maximal operator on the real line for every $1<p<\infty$. It is the unique number $γ_p\in [0,1]$ satisfying $(2p-1)γ_p+(p-1)γ_p^{(2p-1)/(p-1)}=p$. As $p$ goes from $1$ to $+\infty$, $γ_p$ decreases from $1$ to $\sqrt2-1$, and at the endpoints the limiting arguments recover the classical total-variation and Lipschitz inequalities, with the qualitative regularity ingredient at the $BV$ endpoint supplied by Aldaz and Pérez Lázaro.