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Let $D$ be the unit disk and $f: D\rightarrow G$, $\; p_1$ the maximum value of $dist(f(z),f(0))\;$ and $p_2$ the minimum value of $dist(f(z),f(0))$ for $z\in \partial \bar G$
Prove that : $|f(z)-f(0)|\le p_2|z|\;$ and $\; p_1 \le |f'(0)|$

Looks like one has to use Schwarz Lemma but I can not come up with a mapping that works.
Any ideas/tips ?

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Hint: If $$g(z)=\frac{f(z)-f(0)}{p_1}$$ then $g:\Bbb D\to \Bbb D$ and $g(0)=0$.

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