Application of Liouville's Theorem (Complex Analysis)

[Sistine]

Homework Statement

Prove that if $$f$$ is a meromorphic function $$f:\mathbb{C}\rightarrow\mathbb{C}$$ with

$$|f(z)|^5\leq |z|^6\quad\textrm{for all}\quad z\in\mathbb{C}$$

Then $$f(z)=0$$ for all $$z\in\mathbb{C}$$

Homework Equations

Liouville's Theorem

A bounded entire function is constant.

The Attempt at a Solution

I tried applying Liouville's theorem to the quotient $$f(z)^5/z^6$$ which is bounded by 1 but was unsuccessful in proving that f is constant.

 

[JSuarez]

Note that the hypothesis implies that $$f(z)^5/z^6$$ is a bounded function but because of the $$z^6$$ in the denominator, you must prove that it's also entire.

To prove this, note (again, by the hypothesis) that $$f(z)$$ must have a zero at the origin, so either $$f(z)$$ is identically zero, or $$f(z) = z^ng(z)$$; but then

$$f(z)^5/z^6=z^{5n}g(z)^5/z^6$$

And this implies that $$n > 1$$. After simplifying, what can you say about $$g(z)$$?