Suppose \(f\) is analytic in an open set containing the unit disc \(\mathbb{D}\) and \(\abs{f(z)} = 1\) when \(\abs{z} = 1\). Show that either \(f(z) = e^{i\theta}\) for some \(\theta \in \mathbb{R}\) or there are finite number of \(z_k \in \mathbb{D}\), \(k \le n\) and \(\theta \in \mathbb{R}\) such that \[ f(z) = e^{i\theta} \prod_{k=1}^n \frac{z - z_k}{1 - \bar{z}_k z}. \]
The source adds “Also cf. Stein et al, 1.4.7, 3.8.17”, a pointer to exercises in Stein and Shakarchi, Complex Analysis; the statement does not depend on them.