Examples of $f:\mathbb{C} \rightarrow \mathbb{C}$ entire function with $|f(z)| \leq C|z|^2$ $\forall z \in \mathbb{C}$

Question

Find the most general function $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f$ is entire and $\exists C > 0$ with $|f(z)| \leq C|z|^2$ $\forall z \in \mathbb{C}$.

I'm really not sure where to start with this problem.

Answer 1

Hint: Use the Cauchy integral formula on disks $D_R(0)$ and send $R \to \infty$ (this is fine since $f$ is entire). Doing so, you should be able to show that, above some order, the derivatives of $f$ vanish.

Answer 2

Hint: Observe that $f(z)/z^2$ is a bounded, entire function (can you see why it's holomorphic at $0$?)

Answer 3

As $f(z)$ is entire, it can be expressed as a Taylor's series $$f(z)=\sum_{n=0}^\infty a_nz^n$$therefore the most general form comes from $a_n=0$ for $n>2$ and $a_0=a_1=0$ so we have $$f(z)=a_2z^2$$