Quantum effects from particle production on background evolution and
  Cardy-Verlinde formula in f(R) gravity
  We investigate particle production in an expanding universe under the
assumption that the Lagrangian contains the Einstein term $R$ plus a modified
gravity term of the form $R^\alpha$, where $\alpha$ is a constant. Dark fluid
is considered as the main content of the universe and the big rip singularity
appears. Quantum effects due to particle creation is analysed near the
singularity and we find that for $\alpha\in ]1/2, 1[$, quantum effects are
dominant and the big rip may be avoided whereas for $\alpha\in J$ the dark
fluid is dominant and the singularity remains. The Cardy-Verlinde formula is
also introduced and its equivalence with the total entropy of the universe is
checked. It is found that this can always occur in Einstein gravity while in
f(R) gravity, it holds only for $\alpha=\frac{n+1}{2n+6}$, $n$ being the space
dimension, corresponding to the situation in which the big rip cannot be
avoided.
