$\mathbb{C} \setminus f(\mathbb{C})$ contains a disk of radius 1, then $f$ is constant

Question

$f$ is a non constant entire function such that $\mathbb{C}\setminus f(\mathbb{C})$ contains a disk of radius 1, then $f$ is constant.

My thoughts: if we can show $f$ is bounded then by Liouville's theorem we are done. I'm not sure how the given condition implies that.

Answer 1

Call $c$ the center of one such disk, then $z\mapsto\frac1{c-f(z)}$ is an entire bounded function.