Find all Mobius maps $S$ that map unit disk onto itself such that $f[S(z)]=f(z)$ when $|z|<1$.

Question

For $|z|<1$,define $f$ by $$f(z)=\exp\left(-i\log\left[i\left(\frac{1+z}{1-z}\right)\right]^{1/2}\right),$$ where the principal branch of log and the square root are taken.

Find all Mobius maps $S$ that map $B(0,1)$ onto itself such that $f[S(z)]=f(z)$ when $|z|<1$.

My attempt:

I proved that for a Mobius map send unit disk to unit disk, it has to have this form $$S(z)=e^{i\theta}\frac{z-a}{1-\bar{a}z}$$ where $a \in B(0,1)$ But here, we need more conditions to make $f[S(z)]=f(z)$. I guess we need to restrict $a$ to some smaller domain.

Also, I proved that the image of $f$ is the annulus $\{z: 1<|z|<e^{\pi/2}\}$.

Any help?

THANKS!

Answer 1

I'm pretty sure the exact $f$ here is a red herring.

Consider any holomorphic function $f$ defined on the unit disk and ask the same question. Split into two cases, depending on whether the function is constant or not. In the latter, its derivative is nonzero at some point, so you can locally invert it there. What then becomes of your functional equation, and what does it imply about $S$?