  We study an extension of the sandwiched R\'enyi relative entropies for normal
positive functionals on a von Neumann algebra, for parameter values $\alpha\in
[1/2,1)$. This work is intended as a continuation of [A. Jen\v{c}ov\'a, Ann.
Henri Poincar\'e 19, 2513-2542, 2018], where the values $\alpha>1$ were
studied. We use the Araki-Masuda divergences of [Berta et al., Ann. Henri
Poincar\'e 9, 1843-1867, 2018] and treat them in the framework of Kosaki's
noncommutative $L_p$-spaces. Using the variational formula, recently obtained
by F. Hiai, for $\alpha\in [1/2,1)$, we prove the data processing inequality
with respect to positive trace preserving maps and show that for $\alpha\in
(1/2,1)$, equality characterizes sufficiency (reversibility) for any 2-positive
trace preserving map.
