Energy Conditions and Stability in generalized $f(R)$ gravity with
  arbitrary coupling between matter and geometry
  The energy conditions and the Dolgov-Kawasaki criterion in generalized $f(R)$
gravity with arbitrary coupling between matter and geometry are derived in this
paper, which are quite general and can degenerate to the well-known energy
conditions in GR and $f(R)$ gravity with non-minimal coupling and non-coupling
as special cases. In order to get some insight on the meaning of these energy
conditions and the Dolgov- Kawasaki criterion, we apply them to a class of
models in the FRW cosmology and give some corresponding results.
