We give an asymptotically sharp lower bound for the
slope $\lambda (f)$ of a fibration $f:S\longrightarrow B$, where
$S$ is a surface and $B$ is a curve, if there exists an involution
on the general fibre $F$ of $f$. We also construct a new lower
bound of $\lambda (f)$ depending increasingly on the irregularity
of $S$; as an application of this new bound we have a criteria to
control the existence of other fibrations on $S$.