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)?