Sharp derivative inequalities for maximal functions

We determine the optimal constant in the Lᵖ-derivative inequality for the uncentered Hardy--Littlewood maximal operator on the real line for every 1<p<∞. It is the unique number γₚ∈ [0,1] satisfying (2p-1)γₚ+(p-1)γₚ(2p-1)/(p-1)=p. As p goes from $1$ to +∞, γₚ decreases from $1$ to √2-1, 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 P\'erez L\'azaro.