analytic self-map of unit disc

$\displaystyle f:D(0,1) \to D(0,1)$ has a zero of order k at the origin. Show that the function is bounded in magnitude by $\displaystyle \vert z \vert ^k$.

I want to use Schwarz Lemma because the function is analytic on the unit disc, its image is a subset of the unit disc, and f(0)=0. Then I can definitely say that $\displaystyle \vert f(z) \vert \leq \vert z \vert$, but I need more. How might I bring in powers of |z|? Would the other consequence of Schwarz Lemma ($\displaystyle \vert f'(0) \vert \leq 1$) help here?