Slope inequalities for KSB-stable and K-stable families

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 rely on the study of the Harder-Narasimhan filtration, and some generalizations of Castelnuovo's and Noether's inequalities.