TITLE: Proving an entire  function which misses a ball is constant
QUESTION [8 upvotes]: Let $f$ be an entire function s.t $f(\mathbb{C}) \cap B_R(z_0) = \varnothing$ for some $z_0$ and some $R$. Then $f$ is constant.
I guess since the image of the whole plane isn't dense then $f$ doesn't have an essential singularity at infinity. Now I have to exclude it has a polar singularity at infinity..

REPLY [13 votes]: Let $g(z):=\frac 1{f(z)-z_0}$, then $g$ is well-defined and analytic. We have $|g(z)|\leq\frac 1R$ (because $|f(z)-z_0|\geq R$ so $g$ is bounded and hence; by Liouville's theorem, constant. So $f$ is constant.

REPLY [5 votes]: Consider this map: $$z\mapsto\frac{1}{f(z)-z_0}$$