$|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ then $f$ is constant

Question

Let $f\colon\mathbb C \to \mathbb C$ be entire. Show that if $|\operatorname{Im}f(z)|\leq |\operatorname{Re}f(z)|$ for all $z \in \mathbb C$, then $f$ is constant on $\mathbb C$. How I can answer this by considering the distance between $f(z)$ and $i$.

Answer 1

$|\mathrm{Im}f(z)|\le |\mathrm{Re}f(z)|$ implies that $|f(z)-i|\ge \dfrac{\sqrt{2}}{2}$. It follows that $g(z)=\dfrac{1}{f(z)-i}$ is a bounded entire function, and due to Liouville's theorem, it must be a constant.

Answer 2

Alternatively, $\textrm{Re} \, f(z)^2 = \left(\textrm{Re} \, f(z) \right)^2 - \left(\textrm{Im} \, f(z) \right)^2 \geq 0$. This implies that $f(z)^2$ is constant.

Answer 3

follows directly from picard's little theorem