Liouville's theorem problem

Question

Hi i need some hints and help with this problem.

Let $f\in\mathcal O(\mathbb C)$ and assume that $\Re f(z)\geq M$ for all $z\in\mathbb C$. Use Liouville´s theorem to prove that $f$ is constant function.

I am really stuck on this problem.

Answer 1

Assuming $M>0$, we have that $\Re f(z)\geq M$ implies that $|f|\ge M$. In particular, $f$ is never zero. Then $1/f$ is entire and bounded and so constant by Liouville.