TITLE: Liouville's theorem problem
QUESTION [7 upvotes]: 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.

REPLY [9 votes]: 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.