If is a bounded domain which contains origin,
If is analytic with ,
then for any .
Try to solve it by Riemann Mapping Theorem.
Scheme of Proof:
Think about that if is a disk. Then use Schwarz Lemma.
Riemann Mapping Theorem:
If is simple connected domain of , and . Then for any point in , there exists unique function , such that:
- is holomorphic and univalent in .
Thus, if , then , by Schwarz Lemma, with . The equality holds for .
By Riemann Mapping Theorem:
There exists a unique holomorphic and univalent function , such that .
Set , then is holomorphic in and satisfies that , by Schwarz Lemma:
, and .
Thus consider that is univalent, , for any , therefore , for any .