[@Ade:2013lta; @Planck:2015xua]. However, there may also exist some scalar fields at early times of the universe (such as axions [@Peccei:1977hh] or curvaton field [@Lyth:2001nq]), and observations do not exclude the possibilities that the observed CMBR fluctuations come from more than one degree of freedom [@Ade:2013lta; @Planck:2015xua]. Therefore as a generic study, we want to extend the modified gravity inflation models by adding an additional scalar field. Note that Ref. [@Domenech:2015qoa] comes out during the revision of this paper, which studies similar model while taking the scalar field as a curvaton (see also [@Cai:2013caa] for earlier works). Quantum corrections of this type of models have also been studied in e.g. [@Bamba:2014jia].

In usual modified gravity models, it is convinient to transform them from its original Jordan frame to the Einstein frame, where they look like field theory models minimally coupling to gravity, which is easier for the analysis. The transformation is called conformal transformation, of which we change the scaling of the metric while keep the inner constructure unaltered. It has been proved that for pure $f(R)$ modified gravity theories or single scalar field nonminimally coupled to gravity, the curvature perturbations defined in the two frames are equivalent, and one can 