Assume that $f$ and $g$ are analytic and nonvanishing on $B_{2}(0)=$ {$z \in \mathbb{C} :|z|<2$} and that $|f(z)|=|g(z)|$ when $|z|=1$. Can we show that there is an $|a|=1$ such that $f(z)=ag(z)$ for any $z$ in $B_{2}(0)$?
I believe we should look at $f/g$ which would also be analytic. I feel like the $a$ will come out of Schwarz Lemma. Thanks for the help.