How Does Schwarz Lemma Apply to Non-Zero Automorphisms of the Unit Disk?

[asi123]

Homework Statement

Hey guys.

I've been sitting on this one for an hour or so. (got nothing)

http://img138.imageshack.us/img138/765/24869754.png

I think it has something to do with schwarz lemma but I'm not sure.

Any help will be much appreciate.

Thanks a lot.

Homework Equations

The Attempt at a Solution

 

[Gib Z]

It is related to the Schwarz lemma, but its not directly applicable as we are not given f(0)=0. However, the function you have in an automorphism of the unit disk, and there's a way to "bring it back" to Schwarz's lemma.

Theorem: Let $f:D \to D$ be an analytic automorphism of the unit disc and suppose $f(\alpha) = 0$. Then there exists a real number $\theta$ such that:

$$f(z) = \exp(i\theta) \frac{\alpha - z}{1-\overline{\alpha}z}$$

Start of proof:

Let $$g=g_{\alpha}$$ be the above automorphism. Then $$h(w) = f(g^{-1}(w))$$ is an automorphism of the unit disc and maps 0 to 0. It suffices to prove h(w) has the form $$\exp(i\theta) w$$.

To prove that, use the Schwarz Lemma in two different ways.