Actually, I'm solving the following problem. there are some steps I can't understand. Can you guys help me to understand?
The problem is: Find all entire functions that map the unit circle to itself. (problem from Rudin's real & complex analysis chapter 12 ex.4)
excluding the constant function, I first showed that $f$ should have zero inside unit disk and $f$ should map open unit disk into itself by using maximum modulus theorem.
Lots of proof I found then say that its zero should locate at only origin. That's the first part that I cannot understand.
and then they consider $ g(z) = [\bar f(\frac{1}{\bar z})]^{-1} $ and showed that $g(z) = f(z)$ on unit circle. which has a limit point in $C - \{0\}$. so by identity theorem, (that's the second part; I'm not sure $g(z)$ is even analytic except some singular point, since it involves conjugation.) $f(z) = g(z)$ on $C-\{0\}$. Then by considering the order of pole at $0$, we can conclude that.
To study further, I tried to find lots of materials and above discussion may due to identity theorem for meromorphic functions, analytic continuation, etc. But we never learned this.
I think there may be an easier way by just using elementary properties of analytic function or schwarz lemma. Can you help me please?