How can I prove the following assertion?
Let $f$ be a holomorphic function such that $|f|$ is a constant. Then $f$ is constant.
Edit: The more elementary the proof, the better. I'm working my way through a complex analysis workbook, and by this exercise, the only substantial thing covered has been the Cauchy-Riemann equations.
Best Answer
If $|f|$ is constant, then the image of $f$ is a subset of a circle in $\mathbb{C}$. Apply the Open Mapping Theorem.