How to do this directly rather than using Pick's Lemma

Question

In the field of complex analysis, suppose the complex-valued function $w= f(z)$ is a conformal self-map of the open unit disk $\mathbb{D}$. Then in this particular case, we have equality in Pick's Lemma, i.e. $|\frac{dw}{dz}|= \frac{1-|w|^2}{1-|z|^2}$. How can this be computed directly instead of using Pick's Lemma?

Answer 1

Hint: Pick's lemma says that a biholomorphism preserves the Riemannian metric $\frac{2|dz|}{1-|z|^2}$ of the unit disk.

You can show that biholomorphisms of unit disk have the form,

$$e^{i\theta}\frac{z-z_0}{1-\overline{z_0}z},$$

where $z_0 \in D$ and $\theta \in \mathbb{R}.$