Good morning! I have a question on how to show this.
Let $f$ be analytic on a non-empty domain $D$. If $f^2(z) = \overline {f(z)}$ for all $z \in D$ then prove that $f^3$ is constant in $D$. Deduce that $f$ is also constant in $D$.
I'm thinking of proving by contradiction (i. e., $f^3$ is non-constant) and using the open mapping theorem here by considering an open set $U \subseteq D$ which maps into an open map. But I don't know how to move on from here.