Arakelov inequality for families of pairs

We establish an Arakelov-type inequality for a morphism $f\text{\hspace{0.17em}⁣}:(X,\mathrm{\Delta })\to S$, where $(X,\mathrm{\Delta })$ is a simple normal crossing semi-log canonical pair and $S$ is a smooth projective variety. As a consequence, we derive a bound on the Iitaka volumes of algebraic fiber spaces whose geometric generic fiber admits a good minimal model.