We want to show that the function $f$, holomorphic on some domain $D$, is constant in the following cases:
- $z \mapsto \overline{f(z)}$ is holomorphic
- $z \mapsto f(\overline{z})$ is holomorphic
- $|f(z)|$ is constant
Thoughts:
1.
Suppose $g(z)= \overline{f(z)}$ is holomorphic. $f$ is holomorphic so $f$ satisfies the Cauchy-Riemann equations in $D$.
So letting $u=\Re(f), v=\Im(f)$, we have that $u_x=v_y$ and $u_y=-v_x$.
Now $g(z)= \overline{f(z)}$, so $g=u-iv$. Let $w=-v$. Then $u_x=w_y=-v_y$ and $u_y=-w_x=v_x$.
So, $-v_x=v_x$ and so $v$ is a function of $y$ only. However, $v_y=-v_y$ and so $v$ is constant. A similar results holds for $u$ and we are done.
So I think 1. is prove, but I can't see how 2 & 3 work. Help would be appreciated.