  The little rip is a cosmological abrupt event predicted by some phantom dark
energy models that could describe the future evolution of our Universe. This
event can be interpreted as a big rip singularity delayed indefinitely,
although in those models bounded structures will be destroyed in a finite
cosmic time in the future. In this work, we analyse the little rip cosmology
from a classical and quantum point of view within the scheme of alternative
metric $f(R)$ theories of gravity. The quantum analysis is performed in the
framework of $f(R)$ quantum geometrodynamics by means of the modified
Wheeler-DeWitt equation. In this context, we show that the DeWitt criterion can
be satisfied. Similar to what happens in general relativity, this result points
towards the avoidance of the little rip in $f(R)$ quantum cosmology.
