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Motivated by issues on inflation, a generalized modified gravity model isinvestigated, 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 Sitterbackground is examined on-shell as well as off-shell in the Landau gauge. Inaddition, the on-shell quantum equivalence of $f(R)$ gravity in the Jordan andEinstein frames is explicitly demonstrated. Furthermore, we presentapplications related to the stability of the de Sitter solutions and theone-loop quantum correction to inflation in quantum-corrected $R^2$ gravity. Itis shown that for a certain range of parameters, the spectral index of thecurvature perturbations can be consistent with the Planck analysis, but thetensor-to-scalar ratio is smaller than the minimum value within the 1 $\sigma$error range of the BICEP2 result.

One-loop Modified Gravity in de Sitter Universe, Quantum Corrected Inflation, and its Confrontation with the Planck Result