Meromorphic Function is Rational

I'm asking for clarity on the method given above. If there is a duplicate post, it is only duplicate if it addresses this. Thx!

Let $f(z)$ be a meromorphic function on the complex plane, and suppose there is an integer $m$ such that $f^{-1}(w)$ has at most $m$ points for all $w \in \mathbb{C}$. Show that $f(z)$ is a rational function.

A hint/solution has it that we can:

Choose $w_0$ such that the number of points in $f^{-1}(w_0)$ is maximum. Then $f(z)$ attains values $w$ near $w_0$ only near points in $f^{-1}(w_0)$, $\frac{1}{f(z)-w_0}$ is bounded at $\infty$, and $f(z)$ is meromorphic on ${\mathbb{C}^*}$ hence rational.

That $f(z)$ attains values $w$ near $w_0$ only near points in $f^{-1}(w_0)$ is evident by the Opening Mapping theorem, but what of boundedness at infinity? How does this with $f(z)$ being meromorphic make $f(z)$ rational? Are we using Liouville's theorem here?

• I've asked questions about the hint/solution. The posting you've linked does not address those. – Joshua Bunce Mar 2 '16 at 22:05
• Upon further study, I don't think the question I pointed to actually answers your question. To answer the first of your queries, it seems to me that if, for some $\epsilon,\delta>0$, we have $|f(z)-w_0|>\epsilon$ whenever $\mathrm{dist}(z,f^{-1}(w_0))>\delta$, then for large enough $|z|$ we will have $\mathrm{dist}(z,f^{-1}(w_0))>\delta$ and thus $|1/(f(z)-w_0)|<1/\epsilon$. This in turn implies that $1/(f(z)-w_0)$ is meromorphic on $\mathbb{C}^*=\mathbb{C}\cup\{\infty\}$, since there are only a finite number of poles. Not sure of the final conclusion that $f(z)$ is rational. – ForgotALot Mar 3 '16 at 0:19
• Ok, your first statement concerning the distances we can assert by the Open Mapping theorem? Then, that $1/(f(z)-w_0)$ has a removable singularity is a consequence of Riemann's theorem of removable singularities. This in turn implies that a removable singularity is attained on $\mathbb{C}^*$ and thus $1/(f(z)-w_0)$ is meromorphic. – Joshua Bunce Mar 3 '16 at 0:57
• I am unable to convince myself that the open mapping theorem (Rudin Real and Complex Analysis 10.32) implies the statement I made. My statement was just an effort to translate into more precise language the hint "$f(z)$ attains values $w$ near $w_0$ only near points in $f^{−1}(w_0)$." There is a supposed proof of the latter imprecise statement in the question to which I pointed, but I don't understand it. – ForgotALot Mar 3 '16 at 2:35
• Can you remove the duplicate flag? – Joshua Bunce Mar 3 '16 at 7:30