Strict Arakelov inequality for a family of varieties of general type

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 $$\frac{\mathrm{d}\mathrm{e}\mathrm{g}{f}_{\ast }{\omega }_{X\mathrm{/}Y}^{\nu }}{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{f}_{\ast }{\omega }_{X\mathrm{/}Y}^{\nu }}<\frac{n\nu }{2}\cdot \mathrm{d}\mathrm{e}\mathrm{g}{\mathrm{\Omega }}_Y^1(\log S),$$ for all $\nu \in \mathbb{N}$ such that the $\nu$-th pluricanonical linear system $\mathrm{\mid }{\omega }_F^{\nu }\mathrm{\mid }$ is birational. This answers a question asked by M"oller, Viehweg and the third named author.