Show that if $f$ is analytic on a domain $D$, and if $|f|$ is constant, then $f$ is constant.

Question

If $f(z)=0$ for some $z\in D$ then since $0$ is a constant, $f'(z)=0$ on $D$. Also since $f$ is analytic, then by theorem $f(z)$ is constant.

Here is where I get stock!

If $f(z)\not=0$. I want to show that $\operatorname{Re}f$ and $\operatorname{Im}f$ are constant using the hint that $\bar f=|f|^2/f.$

score 2 · Answer 1 · answered Sep 02 '15 at 21:42

2

Hint: Note that $\overline f=\frac cf$ is analytic.

answered Sep 02 '15 at 21:42

Hagen von Eitzen

Thanks! Then since $f$ and $\bar f$ are analytic, it implies $f$ is constant. – hanna Sep 02 '15 at 21:44

1 Answers1