Find all entire functions such that $|f(z)|\leq |z^2-1|$ for all $z\in\mathbb C$. Find all entire functions such that $|f(z)|\leq |z^2-1|$ for all $z\in\mathbb C$.
For large $z$ we have $$|f(z)|\leq 2|z|^2$$ so $f$ is a polynomial of degree $\leq 2$. But how to continue? Could someone give me a hint?

Since $|f(z)|\leq M |z|^2$ we know that $f$ has to be a polynomial of degree $\leq 2$. (Extended version of Liouvilles Theorem).
Furthermore, we have
$$
f(1) = 0 
$$
and
$$
f(-1)=0
$$
so therefore $f(z) = C(z-1)(z+1)=C(z^2-1)$ for some constant $-1\leq C\leq 1$.
