  For $1<p<\infty$ and $M$ the centered Hardy-Littlewood maximal operator on
$\mathbb{R}$, we consider whether there is some $\varepsilon=\varepsilon(p)>0$
such that $\|Mf\|_p\ge (1+\varepsilon)||f||_p$. We prove this for $1<p<2$. For
$2\le p<\infty$, we prove the inequality for indicator functions and for
unimodal functions.
