Describe all holomorphic functions.

Describe all holomorphic functions.

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!