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Let $\mathbb{D}=\{|z|<1,\ z\in\mathbb{C}\}$. Are there any other automorphisms in $\mathbb{D}$ except the Blaschke factor $\displaystyle B_{a}(z)=\frac{z-a}{1-\overline{a}z},\ a\in\mathbb{D}$?

I denote with $\overline{a}$ the complex conjugate of $a$.

Thank you for your time,

Chris

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1 Answer 1

up vote 5 down vote accepted

$f(z) = ze^{i\theta}$ for $\theta \in \mathbb{R}$, the rotations are also conformal automorphism of $\mathbb{D}$.

In fact, if you denote $\Psi_a$ for $a \in \mathbb{D}$ as the Blaschke factor above, you can prove that all conformal automorphism are of the form $f(z) = \Psi_a(z)e^{i\theta}$.

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So, the Blaschke factor is just a particular case for $\theta=0$ and all other possible automorphisms of $\mathbb{D}$ are constructed by multiplying the Blaschke factor with some $e^{i\theta}$ and these are the only ones? –  Chris Jun 17 '12 at 3:03
    
Yes. If I recall the proof has something to do with using the Schwarz Lemma. Look in Stein and Shakarchi Complex Analysis for the proof. –  William Jun 17 '12 at 3:09
    
I think this is in the exercises, not sure which Chapter. –  Kerry Jun 17 '12 at 6:00
    
@ChangweiZhou I believe it is the conformal automorphism chapter. I think it is done in the text where they classify the conformal automorphisms of the disc. –  William Jun 17 '12 at 6:38

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