Show that holomorphic function $f$ is constant if $\Re f$ or $\Im f$ or $|f|$ is constant without using Cauchy Riemann equations

Question

Show that if $f$ is holomorphic in a domain $D$ and if one of the functions $|f|$, $\Re f$, or $\Im f$ is constant on an open subset of $D$, then $f$ is constant on $D$. (without Cauchy Riemann equations, preferably using Open Mapping Principle).

Answer 1

The Open Mapping Theorem states that every non-constant holomorphic function in a domain sends open sets to open sets. Since every domain is open, if $f$ were not constant then $f(D)$ should be open but this cannot be the case if

$\bullet$ $\Re f$ is constant, since in this case $f(D)$ would be included in the vertical line $\{z:\Re z=\Re f\}$ of the complex plane so it couldn't be open.

$\bullet$ $\Im f$ is constant, since in this case $f(D)$ would be included in the horizontal line $\{z:\Im z=\Im f\}$ of the complex plane so it couldn't be open.

$\bullet$ $|f|$ is constant, since in this case $f(D)$ would be included in the circle $\{z:|z|=|f|\}$ of the complex plane so it couldn't be open.