in Conway's Complex Analysis textbook he writes that
if $f$ is entire and $|f|\leq 1+|z|^{1/2}$ then f is constant.
Obviosuly from Liouville's theorem I only need to show that $f$ is always bounded.
For $|z|\leq 1$ it's obvious that $|f|\leq 2$, but how about $|z|>1$?
Thanks in advance.