Proving entire function is identically zero

Question

i am given an entire function $f$, and i know that
$$\lvert f(z)\rvert\leq\sqrt{\lvert z\rvert}\sqrt{\lvert z-1\rvert}$$
for all $z\in\mathbb{C}$. How do i go about proving that $f$ is identically zero?

I had the following idea, clearly $f(0)=0$, and thus if we can prove $f$ is constant, then it is identically zero. To prove it is constant i initially thought i would be able to use liouvilles theorem, however i am unable to rewrite it to be bounded.

(This is not homework, it is a question from a previous exam)

Answer 1

$f(0) = f(1) = 0$ so that $$ f(z) = z(z-1)g(z) $$ with an entire function $g$. Then $$ |g(z)| \le \frac{1}{\sqrt{\lvert z\rvert}\sqrt{\lvert z-1\rvert}} $$ for $z \to \infty$, so that $g$ is bounded, therefore (Liouville!) constant, and consequently identically zero.