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.

Proof:

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

where .

Consider , then

By induction,

, 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.

**Remark:**

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.

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