We give an asymptotically sharp lower bound for the slope $\lambda (f)$ of a fibration $f:S\rightarrow B$, 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 $S$.
"On the slope of fibred surfaces." Nagoya Math. J. 164 103 - 131, 2001.