Consider an $n$-dimensional compact complex manifold with positive first Chern class. Are its Chern numbers $c_i$, for $0 \leq i < n$, bounded in terms of $c_n$?

Remark. The convention $c_0 := 1$, for which the answer is positive since $c_n > 0$, is included to cover the prototypical observation that the Euler number of a hyperbolic surface is negative. In view of this example, I would also like to extend the original question to the realm of log-manifolds and hyperbolicity.