  Fix graphs $F$ and $H$. Let $\mathrm{ex}(n,H,F)$ denote the maximum number of
copies of a graph $H$ in an $n$-vertex $F$-free graph. In this note we will
give a new general supersaturation result for $\mathrm{ex}(n,H,F)$ in the case
when $\chi(H) < \chi(F)$ as well as a new proof of a stability theorem for
$\mathrm{ex}(n,K_r,F)$.
