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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?

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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 $f(z_0)=w$.

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