The following is a problem from Conway chapter 6 section 2: Suppose f is analytic in some region containing $\bar{B}(0;1)$ and $ |f(z)| = 1$ where $|z| = 1$. Find a formula for $f$. (Hint: First consider the case where f has no zeros in B(0;1).)
I have figured that $f$ is analytic over $B(0;1)$, so by Maximum Modulus principle it attains its maximum on the boundary of $B(0;1)$, and so again by the maximum modulus principle $f$ must be constant in $B(0;1)$. So $f(z)=c$ where $|c|=1$ for all $z\in$ $\bar{B}(0;1)$.
Any hints how to continue from here? Also I'm not entirely sure the argument I gave so far is totally correct.