How to prove uniqueness?

Question

For the question asked in Is a Möbius transformation that PRESERVES UNIT DISC uniquely determined by three distinct points and their images? , how can I prove the UNIQUENESS of those transformations (when they exist)?

Thank you in advance!

Answer 1

A Möbius transformation which preserves the unit disk is uniquely determined by the images of two distinct points, i.e. if $T, S$ are automorphisms of the unit disk with $T(z_j) = S(z_j)$ for $z_1 \ne z_2$ then $T = S$.

For a proof consider $\phi = T^{-1} \circ S$. $\phi$ is an automorphism of the unit disk with two fixed points $z_1, z_2$ inside the unit disk. Now there are two possible ways to continue:

• Show that the “mirror points” $1/\,\overline{z_k}$ are also fixed points of $\phi$ and conclude that $\phi$ is the identity.

• Or consider $g = f \circ \phi \circ f^{-1}$ where $f$ is an automorphism of the unit disk with $f(z_1) = 0$. Then $g$ has the two fixed points $0$ and $f(z_2)$. Use the Schwarz Lemma to conclude that $g$ is the identity.