Nevanlinna–Pick interpolation of two sets of points on the unit disk

Question

Given that $z_1, z_2, w_1, w_2 \in D = \{z: |z| < 1\}$ and $\left| \frac{w_1 - w_2}{1 - w_1 \overline{w_2}} \right| \leq \left| \frac{z_1 - z_2}{1 - z_1 \overline{z_2}} \right| $, how do you actually find a holomorphic function such that $f(z_1)= w_1$ and $f(z_2) = w_2$? Is there a way to go trace the proof of the Schwarz-Pick theorem backwards?

Answer 1

What you are probably looking for is a holomorphic function mapping the unit disk $\Bbb D$ into itself with $f(z_1) = w_1$ and $f(z_2) = w_2$. Yes, you can proceed similarly as in the proof of the Schwarz–Pick theorem by "normalizing" one source point and its image to the origin:

$$ T(z) = \frac{z - z_2}{1 - z\overline{z_2} } \quad \text{and} \quad S(w) = \frac{w - w_2}{1 - w\overline{w_2} } $$ are Möbius transformations mapping the unit disk onto itself, and $$ \lambda = \frac{S(w_1)}{T(z_1)} $$ satisfies $\lvert \lambda \rvert \le 1$. Now verify that $$ f(z) = S^{-1}(\lambda T(z)) $$ has the desired properties.