(Solved): 3. The pseudo-hyperbolic distance between two points z,wD is defined by (z,w)=1w ...

3. The pseudo-hyperbolic distance between two points $z,w?\mathbb{D}$ is defined by $?(z,w)=?\frac{z?w}{1?\stackrel{?}{w}z}?.$ (a) Prove that if $f:\mathbb{D}?\mathbb{D}$ is holomorphic, then $?(f(z),f(w))??(z,w)\text{ for all }z,w?\mathbb{D}.$ Moreover, prove that if $f$ is an automorphism of $\mathbb{D}$ then $f$ preserves the pseudo-hyperbolic distance $?(f(z),f(w))=?(z,w)\text{ for all }z,w?\mathbb{D}\text{. }$ [Hint: Consider the automorphism ${?}_{?}(z)=(z??)/(1?\stackrel{?}{?}z)$ and apply the Schwarz lemma to ${?}_{f(w)}?f?{?}_w^{?1}$.] (b) Prove that $\frac{?f^{?}(z)?}{1??f(z){?}^2}?\frac{1}{1??z{?}^2}\text{ for all }z?\mathbb{D}.$ This result is called the Schwarz-Pick lemma. See Problem 3 for an important application of this lemma.