I'm trying to understand this strange exercise in complex analysis:
Let $f : D → \mathbb{C}$ be analytic and $|f(z)| < 1$ for all $z \in D$, and suppose that $f$ has two distinct fixed points $a, b \in D$. Set $T_\alpha(z) := (z − \alpha)/(1 − \bar{\alpha}z)$. By considering the function $T_\alpha \circ f \circ T_{−\alpha}$ with an appropriate $\alpha \in D$, show that $f(z) = z$ for all $z$ in $D$.
What is $T_\alpha$ doing here? Can I accomplish anything by taking that function into consideration? And how can I show that $f$ is the identity map. If I apply Schwarz lemma somehow, I can probably not show anything more than that $f$ is a rotation map, or in other words that $f(z) = \lambda z$ where the absolute value of $\lambda$ is 1.
