  Given a relatively minimal non locally trivial fibred surface f: S->B, the
slope of the fibration is a numerical invariant associated to the fibration. In
this paper we explore how properties of the general fibre of $f$ and global
properties of S influence on the lower bound of the slope. First of all we
obtain lower bounds of the slope when the general fibre is a double cover. We
also obtain a lower bound depending as an increasing function on the relative
irregularity of the fibration, extending previous results of Xiao. We construct
several families of examples to check the assimptotical sharpness of our
bounds.
