  We prove some higher dimensional generalizations of the slope inequality
originally due to G. Xiao, and to M. Cornalba and J. Harris. We give
applications to families of KSB-stable and K-stable pairs, as well as to the
study of the ample cone of the moduli space of KSB-stable varieties. Our proofs
relies on the study of the Harder-Narasimhan filtration, and some
generalizations of Castelnuovo's and Noether's inequalities.
