holomorphic extension of real valued function with bounded derivatives on [0,1]

Question

I have encountered the following question:

let $f \in C^\infty([0,1])$. Assume that there exists some constant number $C>0$, such that for all $ n \in \mathbb{N}$ $$\max_{x \in [0,1]} |f^{(n)}| \leq C^{n+1}\cdot n!$$ prove that there exists some domain $G\subseteq \mathbb{C}$ s.t $[0,1] \subseteq G$, and a function $g$ holomorphic in $G$, such that $g_{|[0,1]}\equiv f$

I've got lost trying to solve this one. I tried to solve for $f$ which is a polynomial and to use Berenstein's theorem, but couldn't solve this case either. Anyone?