  Let $\Omega $ be any set of directions (unit vectors) on the plane. We study
maximal operators defined by \md0 M_\Omega f(x)=\sup_{\delta >0, \omega \in
\Omega} \frac{1}{2\delta}\int_{-\delta}^\delta |f(x+t\omega)|dt. \emd for the
generalized lacunary sets $\Omega $ associated with an integer $\mu >0$. It is
proved the following sharp inequality: $$ \|M_\Omega f(x)\|_2\lesssim
\sqrt{\mu} \|f\|_2. $$
