Blaschke factors are involutive automorphisms of the disk

1. Let \(z, w\) be complex numbers, such that \(\bar{z}w \ne 1\). Prove that \[ \abs{\frac{w - z}{1 - \bar{w}z}} < 1 \quad \text{if } \abs{z} < 1 \text{ and } \abs{w} < 1, \] and also that \[ \abs{\frac{w - z}{1 - \bar{w}z}} = 1 \quad \text{if } \abs{z} = 1 \text{ or } \abs{w} = 1. \]

2. Prove that for fixed \(w\) in the unit disk \(\mathbb{D}\), the mapping \[ F \colon z \mapsto \frac{w - z}{1 - \bar{w}z} \] satisfies the following conditions:

3. \(F\) maps \(\mathbb{D}\) to itself and is holomorphic.

2. \(F\) interchanges \(0\) and \(w\), namely, \(F(0) = w\) and \(F(w) = 0\).

3. \(\abs{F(z)} = 1\) if \(\abs{z} = 1\).

4. \(F \colon \mathbb{D} \to \mathbb{D}\) is bijective.

Hint: Calculate \(F \circ F\).