As we can see from Cartan Theorem can be proved in case by using Schwarz Lemma and Riemann Mapping Theorem. However, this method cannot be applied to .
We shall prove it in a technical way.
Because is a bounded domain, then there exists , such that:
then we have the expansion of as:
assume that the first nonzero is from , then
Consider , then
multiply with and integrate it.
Since , then ,
thus , for any .
where , .
thus , i.e. , which means .
Therefore for all , and .
However this is still for case, for higher dimensional cases, we will talk over it later.
In last post for Cartan Theorem, we required that the domain should be connected, while for this post, we do not have to make this assumption.