# Homework Help: Proof somthing

1. Jul 13, 2010

### asi123

1. The problem statement, all variables and given/known data

Hey guys.

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

http://img138.imageshack.us/img138/765/24869754.png [Broken]

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

Any help will be much appreciate.

Thanks a lot.

2. Relevant equations

3. The attempt at a solution

Last edited by a moderator: May 4, 2017
2. Jul 14, 2010

### 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.

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