Prove that if \(f\) is holomorphic on a connected open set \(\Omega\) and \(f^2(z) = \bar{f(z)}\) then \(f\) is constant.
Solution
Write \(F(z) \da f^3(z)\) so that \[ F(z) = f^2(z) f(z) = \bar{f(z)}f(z) = \abs{f(z)}^2 .\] Since \(F\) is also holomorphic on \(\Omega\) and this calculation shows \(F(\Omega) \subseteq \Omega\), \(F\) must be constant by the open mapping theorem. So \(f^3(z) = c\) for some \(c\), forcing \(f\) to be constant as well.