  Motivated by issues on inflation, a generalized modified gravity model is
investigated, where the model Lagrangian is described by a smooth function
$f(R, K, \phi)$ of the Ricci scalar $R$, the kinetic term $K$ of a scalar field
$\phi$. In particular, the one-loop effective action in the de Sitter
background is examined on-shell as well as off-shell in the Landau gauge. In
addition, the on-shell quantum equivalence of $f(R)$ gravity in the Jordan and
Einstein frames is explicitly demonstrated. Furthermore, we present
applications related to the stability of the de Sitter solutions and the
one-loop quantum correction to inflation in quantum-corrected $R^2$ gravity. It
is shown that for a certain range of parameters, the spectral index of the
curvature perturbations can be consistent with the Planck analysis, but the
tensor-to-scalar ratio is smaller than the minimum value within the 1 $\sigma$
error range of the BICEP2 result.
