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)|<c\\cdot|z|^{n}

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\eq 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)|<c_{1}+c_{2}\\cdot|z|^{n-1}<c^{\\prime}\\cdot|z|^{n-1}

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 $n$ .