  Let $1<p<\infty$. We prove that there exists an $\varepsilon_p>0$ such that
for each $f\in L^p(\mathbb{R})$, the centered Hardy-Littlewood maximal operator
$M$ on $\mathbb{R}$ satisfies the lower bound $\|Mf\|_{L^p(\mathbb{R})}\ge
(1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$.
