On the Lusternik-Schnirelmann category of spaces with 2-dimensional fundamental group

The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical dimension of $X$. We apply this inequality to prove that \cat X\le \lceil\frac{\dim X-1}{2}\rceil+cd(\pi_1(X)) for every complex $X$ with $cd(\pi_1(X))\le 2$.