Biholomorphic Mapping: Proving f(z) = z for All z in Ω?

[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?

 

[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.)

 

[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.