Let $f$ be a non-zero entire function such as $\frac{Re(f)}{Re(f)^2 + Im(f)^2}$ is a bounded function.
Show that $f$ is constant.
I know how to show that $f$ is constant if the real or imaginery part of $f$ is bounded, but I can't see how to implement it here.
Maybe to find a smaller expression that is bounded too?
Any insights?