  We give two kinds of generalizations of Arakelov type inequalities for higher
dimensional families. These results give higher dimensional generalizations (in
both fibers and bases) of the weakly boundedness in Par\v{s}in-Arakelov's
reformulation of the geometric Shafarevich conjecture. As a consequence, we
deduce the following results.
  Hyperbolicity: We give an alternative proof (using the theory of degeneration
of Hodge structure) to the hyperbolicity (in Par\v{s}in-Arakelov's
reformulation, i.e. Viehweg's hyperbolicity conjecture) of the base of a family
with maximal variation whose general fibers admit good minimal models. This has
been proved by Popa-Schnell Hodge theoretically.
  Boundedness: We show the deformation boundedness of admissible families of lc
stable minimal models (introduced by Birkar) with an arbitrary Kodaira
dimension.
