# proof of extended Liouville’s theorem

This is a proof of the second, more general, form of Liouville’s theorem given in the parent (http://planetmath.org/LiouvillesTheorem2) article.

Let $f:\mathbb{C}\to\mathbb{C}$ be a holomorphic function such that

 $|f(z)|

for some $c\in\mathbb{R}$ and for $z\in\mathbb{C}$ with $|z|$ sufficiently large. Consider

 $g(z)=\begin{cases}\frac{f(z)-f(0)}{z}&z\neq 0\\ f^{\prime}(0)&z=0\end{cases}$

Since $f$ is holomorphic, $g$ is as well, and by the bound on $f$, we have

 $|g(z)|

again for $|z|$ sufficiently large.

By induction, $g$ is a polynomial of degree at most $n-1$, and thus $f$ is a polynomial of degree at most $n$.

Title proof of extended Liouville’s theorem ProofOfExtendedLiouvillesTheorem 2013-03-22 16:18:31 2013-03-22 16:18:31 rm50 (10146) rm50 (10146) 8 rm50 (10146) Proof msc 30D20