Existence of biholomorphic map from unit disk to itself that interpolates one set of points

Question

How do you prove that given two points $z_{1}, w_{1} \in D = \{z: |z|<1\} $, there exists a biholomorphic (bijective and analytic) function $f: D \to D$ such that $f(z_{1}) = w_{1}$?

Perhaps using the Schwarz lemma?

Answer 1

Fix a point $w$ in the open unit disk $\mathbb{D}$, and define $$ f_w(z)=\frac{w-z}{1-\overline{w}z}$$ for $z\in \mathbb{D}$. One can show that $f_w$ is a biholomorphic map $\mathbb{D}\to \mathbb{D}$ such that $f_w(0)=w$ and $f_w(w)=0$.

Therefore given two points $w_1,w_2\in \mathbb{D}$, the function $f(z)=(f_{w_2}\circ f_{w_1})(z)$ is a biholomorphic map $\mathbb{D}\to \mathbb{D}$ with $f(w_1)=w_2$.

The maps $f_w$ are sometimes called Blaschke factors, and can be quite useful.