# Biholomorphic mapping

1. Mar 29, 2012

### iamqsqsqs

Suppose f is a biholomorphic mapping from Ω to Ω, if f(a) = a and f'(a) = 1 for some a in Ω, can we prove that f(z) = z for all z in Ω?

2. Mar 29, 2012

### mathwonk

look at the corresponding result for the unit disc, then look at the riemann mapping theorem that says every simply connected proper open set in the plane is equivalent to the disc. (i don't know the answer.)

3. Mar 29, 2012

### micromass

Schwarz lemma ( http://en.wikipedia.org/wiki/Schwarz_lemma ) can be used to prove that the only biholomorphic mappings from the unit disk to itself have the form

$\varphi(z)=\zeta \frac{z-a}{\overline{a}z-1}$

with $|\zeta|=1$ and a in the unit disk. Use this.