Complex Analysis question (maybe Schwarz' lemma?)

Let $$D$$ such that $$\overline{D_0(1)} \subseteq D$$ and let $$f:D\rightarrow\mathbb{C}$$ be a holomorphic function. Assume that $$f$$ holds that $$f(0)=0$$ and $$|f(z)|>1$$ for all $$z \in \partial D_0(1)$$.

Show that $$D_0(1) \subseteq f(D_0(1))$$.

I tried using Schwarz' Lemma but didn't get anywhere. Any idea on how to solve this question?

Let $$w\in D_0(1)$$, then $$|w|<1<|f(z)|$$ for all $$z\in \partial D_0(1)$$. By Rouché's Theorem, $$f$$ and $$f-w$$ have the same number of zeros in $$D_0(1)$$. Since $$f(0)=0$$, there is $$z_0$$ such that $$f(z_0)=w$$.