  Relevant features of the dynamic structure function $S(q,\omega)$ in
$^3$He-$^4$He mixtures at zero temperature are investigated starting from known
properties of the ground state. Sum rules are used to fix rigorous constraints
to the different contributions to $S(q,\omega)$, coming from $^3$He and $^4$He
elementary excitations, as well as to explore the role of the cross term
$S^{(3,4)}(q,\omega)$. Both the low-$q$ (phonon-roton $^4$He excitations and
1p-1h $^3$He excitations) and high-$q$ (deep inelastic scattering) ranges are
discussed.
