I've been stumbling across a couple of questions in Ahlfors that fall along the lines of "If $f$ is an analytic function with property $X$, show that $f$ reduces to a polynomial."
One concrete example would be:
Show that a function which is analytic in the whole plane and has a nonessential singularity at $\infty$ reduces to a polynomial.
I was wondering what the general approach to these problems is. Is it to use some sort of factorization of the zeros/poles of the function and then to use Cauchy's estimate?
My thought process for this particular problem has been as follows:
Consider $g(z) = \frac{1}{f(\frac{1}{z})}$. Then $h(z)=0$ for $z=0$. $h$ cannot vanish identically or else $f$ is identically $\infty$ and then, $f$ does not have a nonessential singularity at $\infty$. Thus, we know that $g(z) = z^h g_h(z)$ where $g_h$ is an analytic function with $g_h(0) \neq 0$. Then we get $\frac{1}{f(\frac{1}{z})} = z^h \frac{1}{f_h(\frac{1}{z})}$ (where $\frac{1}{f_h(\frac{1}{z})} = g_h(z)$), which implies that $f(\frac{1}{z}) = z^{-h} f_h(\frac{1}{z})$. This last expression can also be equivalently written as $f(z) = z^h f_h(z)$. Then, since $f_h(z)$ is bounded as $|z| \to \infty$, we can use Cauchy's estimate to show that the $(h+1)$-th derivative of $f$ is equal to $0$ for all $z$, which implies that the function must reduce to a polynomial.
I'd appreciate it if any of you could tell me if that's correct and/or if there is a better/different way to go about answering the question.
As an extra note, would it be wrong to assume that any approach toward these problems with polynomials is also going to extend in some sense toward problems about rational functions?