Is there a Möbius transformation mapping an arbitrary disk to the unit disk?

Question

I have the following problem. I'm interested in finding a Möbius transformation mapping an arbitrary disk with radius $R$ centered at $z_0$ to $\Bbb{D}$. I thought that it will be something of the form $$T_w(z)=\frac{z-w}{1-w^cz}$$ where $w^c$ is the complex conjugate of $w$.

Now my idea was to replace $w$ with $z_0$ but then I don't think that this works? does it?

Edit I want this for the following reason:

We assume $f$ to be analytic such that $|f(z)|\leq 1$ for $z\in >\Bbb{D}$. Now from the lecture we have the following statement

Let $g$ he an analytic function satisfies $|g(z)|\leq M$ for $|z->z_0|<R$ then if $g$ has a zero of order $m$ at $z_0$ then >$$|g(z)|\leq \frac{M}{R^m}|z-z_0|^m$$ when $|z-z_0|<R$

Using this we need to show that if $f$ has zero of order $m$ at >$z_0\in \Bbb{D}$ then $$|f(0)|\leq |z_0|^m$$

They told us to use a möbiustransformation as above. My Idea was to >transport the $\Bbb{D}$ to the other cirlce $|z-z_0|<R$ becuase then >we can use our statement. But maybe thats completly wrong.

Thanks for your help

Answer 1

Translate the center of your disk to the origin, and then divide by $R$. The composition of these two transformations is what you want.