Q. Let $f$ be an entire function which is not a polynomial. Then prove that the image of the set $\{z \in \Bbb C : |z| \gt 1\}$ under $f$ is dense in $\Bbb C$?
This question had appeared in ISI JRF 2017 exam. I know that the image of a non-constant entire function is dense in $\Bbb C$. Also since the question asks about non-polynomial functions, I think I am missing some facts about polynomial functions. Of course I know that polynomials are entire function.
I found $f(z)=z$ is a polynomial for which the the image of the set $\{z \in \Bbb C : |z| \gt 1\}$ is not dense. So can you provide me some hints for this problem?
Let $g(z)=f\left(\frac1z\right)$. Since $f$ is not polynomial, $g$ has an essential singularity at $0$. So, by the Casorati–Weierstrass theorem, its image is dense. But the image of $g$ is the image of $f$.