Suppose $\mathbb D$ be the unit open disk and $f\in H(\mathbb D)$ and let $u=\operatorname{Re}(f)$, $v=\operatorname{Im}(f)$. Then i need to show that if $|u|+|v|=1$ at every point of $\mathbb D$ then $f$ is constant.
i was trying to apply Rouche's Theorem, but i am not able to get it.
Any type of help will be appreciated. Thanks in advance
If it was not constant, by the open mapping theorem $f(\mathbb{D})$ would be a non-empty open subset of $\mathbb C$. But, since $|u|+|v|=1$,$$f(\mathbb{D})\subset\{x+yi\in\mathbb{C}\,|\,x,y\in\mathbb{R}\wedge|x|+|y|=1\},$$which contains no non-empty open set (it's the square whose vertices are $\pm1$ and $\pm i$).