  Motivated by the ultraviolet complete theory of quantum gravity, for example
the string theory, we investigate a polynomial $f(R )$ inflation model in
detail. We calculate the spectral index and tensor-to-scalar ratio in the $f(R
)$ inflation model with the form of $f(R )=R+{R^2\over 6M^2}+{\lambda_n\over
2n}{R^n\over (3M^2)^{n-1}}$. Compared to Planck 2013, we find that $R^n$ term
should be exponentially suppressed, i.e. $|\lambda_n|\lesssim 10^{-2n+2.6}$.
