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:

, then

.

Proof:

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:

Consider

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 .

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