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Let $f$ be an holomorphic function on a region containing the closure of the disk $D=\{z\in\mathbb{C}:|z|<1\}$, such that $f(0)=0$ and $f'(0)=1$. Then there is a disk $S\subset D$ such that $f$ is injective on $S$ and $f(S)$ contains a disk of radius $\frac{1}{72}$.
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Let $f$ be an holomorphic function on a region containing the closure of the disk $D=\{z\in\mathbb{C}:|z|<1\}$, such that $f(0)=0$ and $f'(0)=1$. Then there is a disk $S\subset D$ such that $f$ is injective on $S$ and $f(S)$ contains a disk of radius $1/72$.
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