  We generalize the classical Szpiro inequality to the case of a semistable
family of hyperelliptic curves. We show that for a semistable symplectic
Lefschetz fibration of hyperelliptic curves of genus $g$, the number $N$ of
non-separating vanishing cycles and the number $D$ of singular fibers satisfy
the inequality $N \leq (4g+2)D$.
