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. 
Jun Wang, Ya-Bo Wu, Yong-Xin Guo, Wei-Qiang Yang, Lei Wang
gr-qc astro-ph.CO