  To study the analog of Suita's conjecture for domains $D \subset
\mathbb{C}^n$, $n \ge 2$, B\l ocki introduced the invariant
$F^k_D(z)=K_D(z)\lambda\big(I^k_D(z)\big)$, where $K_D(z)$ is the Bergman
kernel of $D$ along the diagonal and $\lambda\big(I^k_D(z)\big)$ is the
Lebesgue measure of the Kobayashi indicatrix at the point $z$. In this note, we
study the behaviour of $F^k_D(z)$ (and other similar invariants using different
metrics) on strongly pseudconvex domains and also compute its limiting
behaviour explicitly at certain points of decoupled egg domains in
$\mathbb{C}^2$.
