Liouville's theorem problem
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.
Liouville's theorem problem
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.