I am given that a complex function $f(z) = u(x,y) + iv(x,y)$ is analytic in a region $\Omega$. I am also given a list of conditions and must show that if $f$ satisfies any one of those conditions in $\Omega$, then $f$ is constant in $\Omega$. These are the conditions that I am having trouble with:
a. $f(\Omega) = \{f(z) : z \in \Omega\}$ is a subset of a circle.
b. $u^n(x,y) = v(x,y)$ for some $n \in \mathbb{N}$.
c. $Re(f)$ is analytic on $\Omega$.
My thoughts for part a: is this property saying that f is bounded by the subset of the circle? If so, can I apply Liouville's Theorem and say that since f is bounded, it must be constant?
For part b., my approach was to substitute $u^n(x,y)$ in for $v(x,y)$ in the function, and then use the Cauchy Riemann equations (since $f(z)$ is analytic). Calculating the partial derivatives gave me $u_x = nu^{n-1}u_y$ and $u_y = -nu^{n-1}u_x$. I am not sure if I did the partial differentiation correctly, though, and I'm not sure where to go from here.
For part c., I understand what it means for a complex function to be analytic, but what does it mean for its real part to be analytic? And how does that show that $f$ is constant on the region $\Omega$?
Best Answer
A nice way to prove these results relies solely on the open mapping theorem. One can prove the following, stronger result:
Let $G$ be a $\mathcal{C}^{1}(\mathbb{C},\mathbb{R})$ function such that $0$ is a regular value of $G$ (in particular $G^{-1}(0)$ cannot contain an open set). If $G(f)=0$ on a region $\Omega$, $f$ is constant.
The result is easy to prove: let $f$ be non constant. Then it is an open mapping, which contradicts the regularity of $0$ for $G$ .
This result suffices to prove a&b. For $c$, if $\text{Re}(f)$ is analytic it must be constant by the open mapping theorem, and this implies $\text{Re}(f(\Omega))=\{c\}$ and again by the open mapping theorem we obtain the result.