# Tricky question regarding holomorphy

Question:

Find all holomorphic functions $f(z)$ on $C \setminus \{0\}$ such that

$$f(1) = 1,\ \ \ \ \ \ \ |f(z)| \le \frac{1}{|z|^3}$$

Attempt at solution:

I've discovered that $f(z) = \dfrac{1}{z^3}$ works. And that other monomials $f(z) =z^n \ (n \ne 3)$ don't work. But can't get a general result using power series expansion of $f$.

Also tried to use various Complex Analysis theorems such as Schwarz Lemma, Maximum Modulus Principle, etc. to rule out cases, but not successful.

As such, any help would be much appreciated. Thank you.

Hint: Consider $g(z)=z^3f(z)$, and show that it has a holomorphic extension to $\mathbf C$. What can you say about that extension? Conclude.
• @ConanWong Why would you have $g(0)=0$? I was thinking more along the lines that $g$ is holomorphic on $\mathbf C$ and bounded. – jathd Dec 4 '12 at 1:25