  We investigate the cosmological perturbations in generalized gravity, where
the Ricci scalar and a scalar field are non-minimally coupled via an arbitrary
function. In the Friedmann-Lemaitre-Robertson-Walker background, by studying
the linear perturbation theory, we separate the scalar type perturbations into
the curvature perturbation and the entropy perturbation, whose evolution
equations are derived. Then we apply this framework to inflation. We consider
the generalized slow-roll conditions and the quantization initial condition.
Under these conditions, two special examples are studied analytically. One
example is the case with no entropy perturbation. The other example is a model
with the entropy perturbation large initially but decaying significantly after
crossing the horizon.
