Schwarz-Pick lemma for disk

Statement

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

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

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

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

Facts used

• Schwarz lemma

Proof

Fill this in later