I have a conjecture that I can´t prove nor disprove, any help on doing so will be very grateful.
Let $f: \{z: |z|<2\} \to \mathbb C$ be a non constant analytic function such that if $|z|=1$ then $|f(z)|=1$.
Is it true that the zeros of $f$ can not be in $\{ z: 1/2< |z| < 2 \}$ ?
I have successfully proven, by the maximum modulus theorem, that $f$ must have a zero inside $\{ z: |z|<1 \}$. However, I can not seem to prove that all the zeros must be in $\{ z:|z| < 1/2 \}$, neither to find a counter example.
It is true. If a function is unimodular on the unit circle ($|f(z)|=1$), you can use Schwarz reflection principle that tells you that the zeros and poles are symmetric wrt the unit circle ($z\ \leftrightarrow\ \bar{z}^{-1}$). There are no poles inside $|z|=1$, so there are no zeros outside $|z|=1$ for the analytical continuation. All poles are outside $|z|=2$ then all zeros are inside $|z|=1/2$.