TITLE: Describe all holomorphic functions.
QUESTION [6 upvotes]: Problem: Describe the class of all holomorphic functions on $\mathbb{C}-\{0\}$ such that 
$$\sup_{(x,y)\neq (0,0)}\frac{|f(x+iy)|}{|\log(x^2+y^2)|}<\infty.$$
Attempt at a solution:
Let $z=x+iy$, then we have:
$$\frac{|f(z)|}{|\log|z|^2|}\leq c$$.
 So,
$|f(z)|\leq c|\log|z|^2|.$
For large enough $z$, we have $|\log|z|^2|\leq |z|^2$ so we get:
$$|f(z)|\leq c|z|^2.$$ Now by extented Liouville's Theorem, $f(z)$ must reduce to a polynomial of degree at most two. 
Is this correct?
Thanks!

REPLY [6 votes]: We also need to consider the fact that $f$ might have a singularity at $z=0$. However, we can handle that because
$$
\begin{align}
\lim_{z\to0}|zf(z)|
&\le\lim_{z\to0}|z|2c\log(|z|)\\
&=0
\end{align}
$$
Riemann's Theorem says that the singularity at $z=0$ is removable. Now, using Cauchy's Integral Formula we get that
$$
f^{(n)}(z)=\frac{n!}{2\pi i}\oint\frac{f(w)\,\mathrm{d}w}{(w-z)^{n+1}}
$$
where the integral is over a counter-clockwise circle of radius $R$. This means that
$$
\left|\,f^{(n)}(z)\,\right|\le\frac{n!}{R^n}2c\log(R)
$$
Just as with the proof of Liouville's Theorem, we show that $f^{(n)}(z)=0$ and $f$ must be constant, and since $f(z)=0$ when $|z|=1$, we know that $f(z)=0$.

Note added
I mentioned above that your conditions imply that $f(z)=0$ when $|z|=1$. This in itself implies that $f(z)=0$ everywhere since $\{z:f(z)=0\}$ has an accumulation point (see this section).