TITLE: $f:\mathbb{C}\rightarrow\mathbb{C}$ is entire function such that $g(z)=f(1/z)$ has a pole at $z=0$, then is $f$ surjective?
QUESTION [6 upvotes]: $f:\mathbb{C}\rightarrow\mathbb{C}$ is entire function such that $g(z)=f(1/z)$ has a pole at $z=0$, then is $f$  surjective?
I can prove that $f$ will be a polynomial. and hence  $f$ is surjective. am I right?

REPLY [3 votes]: Yes, since your polynomial must have degree at least 1, this follows from the fundamental theorem of algebra.
(The fact that $g$ is a polynomial is proabably easiest to see from the Taylor series expansion of $f$ which immediately gives you a Laurent series expansion for $g$, but this is probably what you did already.)