  The effective inflationary Lagrangian is a prominent challenge for
theoretical cosmologists, since it may contain imprints of the quantum epoch.
In view of the fact that higher order curvature terms might be present in the
effective inflationary Lagrangian, in this work we introduce the theoretical
framework of power-law $f(R)$ gravity corrected canonical scalar field
inflation, aiming to study the inflationary dynamics of this new framework. The
main characteristic of this new theoretical framework is the dominance of a
power-law $f(R)$ gravity term $\sim R^n$, with $1<n<2$, compared to the
Einstein-Hilbert term $\sim R$. In effect, the field equations are controlled
by the $R^n$ term in contrast to the Einstein-Hilbert canonical scalar theory.
We extract the slow-roll field equations and we calculate the slow-roll indices
of the resulting theory which acquire quite elegant final form, when the
slow-roll conditions hold true. Accordingly, we examine quantitatively the
inflationary phenomenological implications of the theoretical framework we
introduced, by choosing simple hybrid-like scalar field potentials. As we
evince the resulting theory is in good agreement with the latest Planck data
for a wide range of the free parameters of the model. Thus our theoretical
framework makes possible to obtain viable inflationary theories, which
otherwise would be non-viable, such as the simple power-law $f(R)$ gravity
model or the simple power-law scalar model.
