The pseudo-hyperbolic distance between two points \(z, w \in \mathbb{D}\) is defined by: \[\rho(z, w) = \left| \frac{z - w}{1 - \bar{w} z} \right|.\]
Prove that if \(f: \mathbb{D} \to \mathbb{D}\) is holomorphic, then: \[\rho(f(z), f(w)) \le \rho(z, w) \quad \text{for all } z, w \in \mathbb{D}.\] Moreover, prove that if \(f \in \operatorname{Aut}(\mathbb{D})\) is a conformal automorphism of \(\mathbb{D}\), then \(f\) preserves the pseudo-hyperbolic distance: \[\rho(f(z), f(w)) = \rho(z, w) \quad \text{for all } z, w \in \mathbb{D}.\]
Prove the Schwarz-Pick Lemma: for any holomorphic function \(f: \mathbb{D} \to \mathbb{D}\), \[\frac{|f'(z)|}{1 - |f(z)|^2} \le \frac{1}{1 - |z|^2} \quad \text{for all } z \in \mathbb{D}.\]
Solution
For \(a\in\mathbb D\), define the disk automorphism \[ \phi_a(\zeta)=\frac{\zeta-a}{1-\overline a\zeta}. \] Then \[ \rho(z,w)=|\phi_w(z)|. \]
Fix \(w\in\mathbb D\) and set \[ g=\phi_{f(w)}\circ f\circ\phi_w^{-1}. \] Then \(g:\mathbb D\to\mathbb D\) is holomorphic and \(g(0)=0\). Schwarz’s lemma gives \[ |g(\zeta)|\le|\zeta|. \] Taking \(\zeta=\phi_w(z)\) yields \[ \rho(f(z),f(w)) =|\phi_{f(w)}(f(z))| \le|\phi_w(z)| =\rho(z,w). \] If \(f\) is an automorphism, apply the same inequality to \(f^{-1}\) and the pair \(f(z),f(w)\) to obtain the reverse inequality. Hence automorphisms preserve \(\rho\) exactly.
For the infinitesimal form, Schwarz’s lemma also gives \(|g'(0)|\le1\). Since \[ \phi_a'(a)=\frac1{1-|a|^2} \] and, because \(\phi_w^{-1}(\zeta)=(\zeta+w)/(1+\overline w\zeta)\), \[ (\phi_w^{-1})'(0)=1-|w|^2, \] the chain rule gives \[ g'(0) = \frac{1-|w|^2}{1-|f(w)|^2}f'(w). \] Therefore \[ \frac{|f'(w)|}{1-|f(w)|^2} \le \frac1{1-|w|^2}. \] Renaming \(w\) as \(z\) gives the Schwarz-Pick inequality.