Let $f$ be an holomorphic function from $\mathbb{D}$ to $\mathbb{D}$ and let $z \in \mathbb{D}$. I have to prove that $$ \frac{|f(0)| - |z|}{1 + |f(0)| |z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1 - |f(0)| |z|} $$ and I have thought to use Schwarz-Pick theorem but I haven't obtain any result. Any help? Thank you very much.
Let $D := \{z \in \mathbb{C}: |z| &lt; 1\}$ and $f\colon D \rightarrow \mathbb{C}$ be holomorphic. Suppose $\lvert f(z)\rvert \leq 1$ on $D$, show that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1 - |f(0)||z|} \ \forall z \in D.$$  I have tried using Cauchy Integral Formula to $f$ and expanding $f$ at $0$, but I have no idea why $f(0)$ appears in the inequality in the former case. Both answers and hints are appreciated and thanks in advance!
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