What kind of singularity an entire function $f$ have at infinity provided $f^{-1}(B_{1})$ is bounded , where $B_{r}$ is a closed disk $\{z\in \mathbb{C}:|z| \leq r\}$
Solution i tried-First i suppose an example $f(z)=z$, which satisfies all the given condiitons and it have a Pole of order $1$ at infinity
But there can be other possibility , i'm not getting how to get that,
someone recommend me to use "Casorati–Weierstrass" theorem , but can see how can that help in inverse of a entire function
Please Help
Thank you
Best Answer
It can only have a pole at $\infty$ unless it is a constant $c$ with $|c| >r$. If it has an essential singularity then, in every neighborhood of $\infty$, the function would assume values inside $B_1$ (by Casorati–Weierstrass theorem). If it is a removable singularity then it is a constant by Liouville's Theoem. It should also be noted that if $f$ is entire and $f$ has a pole at $\infty$ then it is a polynomial.