I need help with the problem 6 of the chapter VI section 2 from the book "Functions of One Complex Variable" by John B. Conway.
The problem ask for find a formula for a supposed function $f$ that is analytic in some region that contains the closed unit disk such that $|f(z)| = 1$ for $|z| = 1$.
Assuming that $f$ have no zeros in the open disk I can see that $f$ has to be some constant of module one in the closed disk (Applying Maximum Modulus Theorem to $f$ and $1/f$).
But I don't know how to extend that to all the region or what I can do when the function have zeros in the open disk.