Let $G\subset \mathbb{C}$ be a region, $f$ a holomorphic function.
Then it does hold that:
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If $f(G) \subset \mathbb{R} \Rightarrow f$ is constant
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If $|f(z)|=1$ for all $z \in G$, then $f$ is constant.
Proof of 1:
$f$ is holomorphic, so it holds that $$u_x=v_y; u_y=-v_x$$ where $f(z) = u(x,y)+iv(x,y)$, then $u_x=0$ and $u_y=0$ so $u,v$ both must be constant
Proof of 2:
$$|f(z)|= 1 \Rightarrow u^2+v^2=1 $$
taking the partial derivatives:
$$2uu_x+2vv_x=0 , 2uu_y+2vv_y= 0$$
putting inside the Cauchy-Riemann equations:
$$uu_x-vu_y=(u^2u_x^2-2uvu_xu_y+v^2u_y^2)=0; uu_y+vu_x=(u^2u_y^2+2uvu_xu_y+v^2v_y^2)=0$$
so either $u=v = 0$ or $u_x=u_y=0$
so $$\Leftrightarrow (u^2+v^2)(u_x^2+u_y^2)=0.$$
If everything was done correctly so far, wouldn't
switching $1$ with a constant $M$ give a proof for Liouville's theorem?
Best Answer
Both claims follow at once from the fact that a non-constant holomorphic function is an open map: Apparently, the image of $f$ is not open in $\mathbb C$.