The Bogomolov-Miyaoka-Yau inequality for compact complex manifolds with ample canonical bundle and the Kobayashi-Lubke inequality for holomorphic stable vector bundles involve the first two Chern classes.

Naively, I would expect that a more involved stability condition (corresponding to solving a complicated PDE akin to the Hermite-Einstein/Kahler-Einstein equations) would lead to an inequality involving the higher Chern classes. Is there a reason why this has not been studied ? Even a meta reason as to why such universal inequalities are not expected would be nice.