Find the entire functions that satisfy the following condition

Question

the problem at hand is to find all the entire functions satisfying $|f(z)| = |z|^2$ for all $z \in \mathbb{C}$

My attempt:
$f$ grows at most as a polynomial of degree 2 therefore, it is a polynomial of degree 2.
Then $f = a_0 + a_1z +a_2z^2$

I'm sure i'm solving it correctly.

Any help would be awesome

Answer 1

Notice that $z^{-2}f(z):\Bbb C\setminus\{0\}\to\Bbb C$ is bounded and analytic. Therefore, it extends to an entire function $\Bbb C\to\Bbb C$. Which must be constant by Liouville's theorem.