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"We saw in class that a mobius transformation can have at most one fixed point (or else is the identity map), extend this idea to all analytic functions mapping from a domain[itex]\subset[/itex]ℂ to itself."

I have no idea what to do. I totally understand the mobius version of this, but it is way easier. I saw a similar problem online for the unit disk, but then you have Schwart's Lemma. I also had a couple other thoughts that led nowhere. |f(z)-z| attains a minimum at any fixed point. Rouche's theorem would be nice if we could say something about curves, but I don't know how to extend that to an arbitrary domain.

Any push in the right direction would be appreciated.