Complex Analysis

Suppose that $f$ is a holomorphic function in $\mathbb{C}\setminus\{0\}$ and satisfies $$\vert f(z) \vert \leq \vert z \vert ^{2} +\frac{1}{\vert z \vert ^{2}}$$

If f is an odd function, find an expression for f.

I thought about calculating the Laurent Serie of f, but I can't calculate Laurent's series explicitly (there is no way to calculate the coefficients), Dai thought about estimating, using the invariance of the circle to know which one cancels out. I cannot develop this question