Fixed Points of analytic functions

[develmath]

Hi, I am in honors track Complex Analysis, and I think I've reached my limit. We got this proof, and I don't know where to start.

"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\subsetℂ 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.

 

[develmath]

If you totally understand the mobius part, then why does (z+2)/z have two fixed points and it's not the identity map?

Two things:
1. Sorry, I was stupidly thinking while I was writing and got my stuff wrong. A mobius transformation has two fixed points (I "understand" in the sense that I can see that they can be solved for algebraically). I accidentally said "at most one" because I was thinking that that is the appropriate extension of the problem (analytic has at most one fixed point). I feel like Rouche implies that it is at most one fixed point for f(z)-z in this scenario, but can't figure out how to make it work.

2. I also had a problem with this statement at first and generated a similar polynomial counter-example, but I **think** the critical thing is that the domain is not allowed to be ℂ (strict subset is used), and I can't construct a counterexample with that restriction. So for example, if I had my domain being ℂ-{0}, your mobius transformation is certainly still well defined, but it maps -2 to 0 which would be out of the range. I could remove -2 from the domain, but then it comes out of the range again, so I still have a problem. So I can't construct a counterexample to the problem.

 

[Dick]

I think you are right. It has to map a proper domain of C to the same proper domain. That's the part I wasn't really thinking about, so I deleted my post. I was just fixating on the fact a mobius transformation generally has two fixed points.

 

[develmath]

Dah! Checking over what I wrote, found I left off "simply connected" which is of course critical.