Properties of the eigenvalues of $A \cdot B$

Question

Take two (real) matrices $A$ and $B$, where both have (real) eigenvalues within the unit circle, $B$ is also a diagonal matrix.

Can I say something about the eigenvalues of the product $A \cdot B$ ?

I suspect that the eigenvalues of $A \cdot B$ are also within the unit circle. Any help/reference for a proof of that?

Thanks