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 fiber $F$ of general type. We show the strict Arakelov inequality

$$\frac{\mathrm{deg}{f}_{\ast }{\omega }_{X/Y}^{\nu }}{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{f}_{\ast }{\omega }_{X/Y}^{\nu }}<\frac{n\nu }{2}\cdot \mathrm{deg}{\mathrm{\Omega }}_Y^1(\log S),$$

for all $\nu \in \mathbb{N}$ such that the $\nu$-th pluricanonical linear system $|{\omega }_F^{\nu }|$ is birational. This answers a question asked by Möller, Viehweg and the third named author ^[1].