Let $f$ be an entire function on $\mathbb{C}$ such that (a) $f$ has no zeros, and (b) $f^{-1}(1)$ is finite. How can I prove that $f$ is constant?
I was thinking of using Little Picard, but this gets me no further than knowing that $f$ has image that is all of $\mathbb{C}^{*}$. Thank you!
$f$ is entire and omits $0$ as a value, so it can be written as $f(z)=e^{g(z)}$ for some entire function $g$. Next, $f^{-1}(\{1\})$ being finite implies that $g$ omits infinitely many values in $2\pi i \Bbb{Z}$, so $g$ must be constant by Picard, hence so is $f$.