Consider the function on $\mathbb R$ defined by
$f(x)=\begin{cases}\frac{1}{|x|\left(\log\frac{1}{|x|}\right)^2} & |x|\le \frac{1}{2}\\ 0 & \text{otherwise}\end{cases}$
Now suppose $f^*$ is the maximal function of $f$, then I want to show the inequality $f^*(x)\ge \frac{c}{|x|\left(\log\frac{1}{|x|}\right)}$ holds for some $c>0$ and all $|x|\le \frac{1}{2}$.
But I don't know how to prove it. Can anyone give me some hints?
Thanks very much.