# existence of non constant entire function that is bounded outside an annulus

Is there a function $f$ which is not constant, analytic in $\mathbb{C}\setminus\{0\}$ and

$$\forall z\in\{w\in\mathbb{C}: 0<|w|<\frac{1}{100} \lor |w|>100\}:|f(z)|<800$$

Probably not.

Such $f$ is bounded in a punctured neighborhood of $0$, therefore $0$ is a removable singularity of $f$. So, we can assume that $f$ is entire.

No, any such function must be constant. Indeed, as you note your function must be entire, so if $A$ denotes your (open) annulus, $f$ is bounded outside of $A$ and $f$ is bounded on the closure of $A$ by continuity and compacity of the closure of $A$. So by Liouville's theorem, it must be constant.
• I forgot about the compacity of $A$. Thank you! – user554578 Jun 15 '18 at 15:17