rouche's theorem without strict inequality

Question

I've come across a problem involving Rouche's theorem. It asks whether we can say something about the roots of $f(z)=z$ if we know that on the boundary $ \mid z \mid = 1$ we have $\mid f(z) \mid \leq 1$. If the inequality is strict, this is easily solved but I don't know whether we can say something in this case.

Answer 1

If $f(z) \equiv z$ then every point in the disk is a root and if $f(z)=z^{2}$ then $0$ is the only root in the open disk. So we cannot say anything about the number of roots in this case.