$ r\in\mathbb{R}_{>0} $. Let $ f,g:\overline{U_r(0)}\rightarrow\mathbb{C} $ be continuous and without zero. Let $ f,g $ be holomorphic on $ U_r(0) $ and $ |f(z)|=|g(z)| $ for all $ z\in\partial U_r(0) $. Show that there is a $ \lambda\in\mathbb{C} $ with $ |\lambda|=1 $, such that $ f=\lambda g $.
I am trying to solve this and I have seen a hint that applying the maximum modulus principle to $f/g$ implies $f/g$ constant and thereby solves the exercise. But I fail to apply. I guess it's about this version of the maximum modulus principle:
If $D$ is a bounded domain and $f$ is holomorphic on $D$ and continuous on its closure $\bar D$, then $|f|$ attains its maximum on the boundary $∂D := D \backslash \bar D$.
This tells us only that $f/g$ attains its maximum on $\partial U_r(0)$, which we know to be $1$. So $|(f/g)(z)| \leq 1$ for all $z \in U_r(0)$. But how do we know that $f/g$ is constant?
Best Answer
You have already solved the exercise in the comments, I present a solution which is quite similar.
Let $f_1 = f/g$, $f_2 = g/f$ both of these functions are holomorphic and thus will attain maxima at the boundary. However both of them are $=1$ on the boundary thus.
$$|f_1| \le 1 \implies |f| \le |g|$$ $$|f_2| \le 1 \implies |g| \le |f|$$
Thus we have $|f/g| = 1$