what is the mobius transformation that maps a circle with center $z_0$ and radius $R$ to the unit circle?
I want to find the mobius function $f(z)$ that transforms the circle with center $z_0$ and radius $R$ to the unit circle centered at the origin.
There are no one answer, because you only said the transformation of domain.
For all $\alpha\in\Bbb C\ (|\alpha|=1)$, $f:z\mapsto\frac\alpha R(z-z_0)$ could be answer. Why?
If $|z-z_0|<R$, then $|f(z)|=\left|\frac{\alpha}{R}(z-z_0)\right|=\left|\frac{\alpha}{R}\right||z-z_0|<\frac{|\alpha|}RR=1$. $f(z)$ is element of unit disk.
With the same logic, showing $f$ maps your circle to unit circle is not so hard.