  In this work, we provide a strengthening of the data processing inequality
for the relative entropy introduced by Belavkin and Staszewski (BS-entropy).
This extends previous results by Carlen and Vershynina for the relative entropy
and other standard $f$-divergences. To this end, we provide two new equivalent
conditions for the equality case of the data processing inequality for the
BS-entropy. Subsequently, we extend our result to a larger class of maximal
$f$-divergences. Here, we first focus on quantum channels which are conditional
expectations onto subalgebras and use the Stinespring dilation to lift our
results to arbitrary quantum channels.
