Conformal maps from the upper half-plane onto the disc

For \(\beta\in\mathbb H\), define \[ \phi_\beta(z)=\frac{z-\beta}{z-\overline\beta}. \] For \(z\in\mathbb H\), \[ |z-\beta|<|z-\overline\beta|, \] so \(|\phi_\beta(z)|<1\). The map \(\phi_\beta\) is Möbius, sends \(\beta\) to \(0\), and maps the real axis to the unit circle; hence it is a conformal bijection \(\mathbb H\to\mathbb D\).

Now let \(F:\mathbb H\to\mathbb D\) be any conformal bijection and let \[ \beta=F^{-1}(0). \] Then \[ G=F\circ\phi_\beta^{-1}:\mathbb D\to\mathbb D \] is an automorphism fixing \(0\). By Schwarz’s lemma applied to \(G\) and \(G^{-1}\), \[ G(w)=e^{i\theta}w \] for some \(\theta\in\mathbb R\). Therefore \[ F(z)=e^{i\theta}\phi_\beta(z) =e^{i\theta}\frac{z-\beta}{z-\overline\beta}. \] Conversely, every map of this form is a rotation composed with the conformal bijection \(\phi_\beta\), so it maps \(\mathbb H\) conformally onto \(\mathbb D\).