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 Letter, 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.