How do I find an upper bound on $z/(z^3+1)$ on a circular path with radius R centred at the origin?

Question

I want to use the estimation theorem,

so I want to find an $M$ such that $|\frac{z}{z^3 + 1}| < M$

I cant seem to work with the $z^3$.

$$|z^3 + 1| \geq |z^3 - 1| $$ is just not true.

How can I do this?