# How to apply maximum modulus principle?

$$r\in\mathbb{R}_{>0}$$. 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?

• Note that you can apply the MMP to both $f/g$ and to $g/f$ ... Jul 11, 2023 at 9:19
• Do you also know that a nonconstant holomorphic functions attains its maximum only on the boundary? Jul 11, 2023 at 9:23
• @MartinR thanks! I think I got it: Suppose $|(f/g)(z)| = c <1$ for some $z \in U_r(0)$. But then $(g/f)(z)=1/c>1$ which is a contradiction since we can apply MMP to $g/f$ as well. So we get $(f/g)(z)=1$ for all $z\in U_r(0)$. So it takes its maximum in $U_r(0)$, so by applying MMP again (the "normal form" this time), we get that $f/g$ is constant, which solves the exercise. Is this correct? Jul 11, 2023 at 9:39
• Almost. It should be $|(f/g)(z)|=1$ etc. Jul 11, 2023 at 10:57

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 $$|f/g| = 1$$