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?
Best Answer
Hint: $f(z) \overline{f(1/\overline{z})}$ is analytic on ${\mathbb C} \backslash \{0\}$. What is it on the unit circle?