Schwarz-Pick lemma for disk

Statement

Let $D$ denote the open unit disc in $\mathbb{C}$. Consider the function $\mathrm{\Delta }:D\times D\to \mathbb{R}$, given by:

$\mathrm{\Delta }(z,w):=\frac{|z-w|}{|\overline{w}z-1|}$

Then, if $f:D\to D$ is a holomorphic function, we have:

• For all $z,w\in D$, $\mathrm{\Delta }(f(z),f(w))\le \mathrm{\Delta }(z,w)$
• $f$ is a conformal automorphism of $D$ if and only if for all $z,w\in D$, we have $\mathrm{\Delta }(f(z),f(w))=\mathrm{\Delta }(z,w)$

Facts used

• Schwarz lemma

Proof

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