Automorphisms of the unit disc as rotated Blaschke factors

For \(a\in\DD\) let \(\psi_a(z)\coloneqq\dfrac{a-z}{1-\overline{a}z}\) be the Blaschke factor at \(a\). The group \(\Aut_\CC(\DD)\) of biholomorphisms \(\DD\to\DD\) is \[ \Aut_\CC(\DD)=\{\lambda\psi_a : a\in\DD,\ \lambda\in S^1\}, \] so every automorphism of \(\DD\) is a Blaschke factor followed by a rotation [@SS03].