  In this short note, we give an alternative proof of the semipositivity of the
Chow-Mumford line bundle for families of K-semistable log-Fano pairs, and of
the nefness threeshold for the log-anti-canonical line bundle on families of
K-stable log Fano pairs. We also prove a bound on the multiplicity of fibers
for families of K-semistable log Fano varieties, which to the best of our
knowledge is new.
