Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a non-constant entire function. Is it true that for all $a,b > 0$, there exists $s \in \mathbb{C}$ such that $|s|>a$ and $|f(s)|>b$.
I have picard's theorem in mind, but it only says about $f(s)$ , i am not sure how to prove the part $|s|>a$.
Any ideas how to prove or disprove ?
Thanks
Best Answer
It is true. Otherwise there would be numbers $a, b > 0$ such that $|f(z)| \le b$ for all $z$ with $|z| > a$. That would imply that $f$ is bounded by $$ M = \max(b, \, \max_{|z| \le a } |f(z)| ) $$ in contradiction to Liouville's theorem (and also contradicting Picard's little theorem).