  We establish a lower bound for the Donaldson-Futaki invariant of optimal
degenerations produced by the K\"ahler-Ricci flow in terms of the greatest
Ricci lower bound on arbitrary Fano manifolds. As an application, we can
generalize the finiteness of the Futaki invariants on K\"ahler-Ricci solitons
obtained by Guo-Phong-Song-Sturm to the space of all Fano manifolds. Also, we
discuss the relation to Hisamoto's inequality for the infimum of the
$H$-functional.
