  Let $f:\, X\to Y$ be a semistable non-isotrivial family of $n$-folds over a
smooth projective curve with discriminant locus $S \subseteq Y$ and with
general fibre $F$ of general type. We show the strict Arakelov inequality
\[\frac{\mathrm{deg}\, f_*\omega_{X/Y}^\nu}{\mathrm{rank}\,
f_*\omega_{X/Y}^\nu} < {n\nu\over 2}\cdot\mathrm{deg}\,\Omega^1_Y(\log S),\]
for all $\nu\in \mathbb N$ such that the $\nu$-th pluricanonical linear system
$|\omega^\nu_F|$ is birational. This answers a question asked by M\"oller,
Viehweg and the third named author.
