Schwarz lemma as a dichotomy

Let \(f\colon\DD\to\DD\) be holomorphic with \(f(0)=0\). Then exactly one of the following holds.

1. There is \(\theta\in\RR\) with \(f(z)=e^{i\theta}z\) for all \(z\in\DD\).

2. \(\abs{f'(0)}<1\), and \(\abs{f(z)}<\abs{z}\) for all \(z\in\DD\setminus\{0\}\).