We have seen from the post that about the Cartan Theorem, with the univalent funtion’s Laurent expansion as:
where , which is the unit disk.
We call the family of univalent functions with above expansion is family.
First, we prove that if the normalized function:
Consider , then
is nonzero in and is an even function.
Thus we can choose one branch of such that . Also we have
is an even function, consider , then is odd and univalent in .
Consider the expansion of , we have
, thus .
Assume that , then
, where .
Because is univalent, then we have:
maps into a close Jordan curve in complex plane. Consider the area then:
Let’s prove the Koebe Theorem:
which is univalent in , which maps .
Then satisfies the theorem.
Now we start with our function , which is univalent in unit disk , and .
then think about , where . Then we have that , which means .
And use the clue that . We have .