  Generalizing recent results of Egge and Mongelli, we show that each diagonal
sequence of the Jacobi-Stirling numbers $\js(n,k;z)$ and $\JS(n,k;z)$ is a
P\'olya frequency sequence if and only if $z\in [-1, 1]$ and study the
$z$-total positivity properties of these numbers. Moreover, the polynomial
sequences $$\biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and}
\quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0}$$ are proved to be
strongly $\{z,y\}$-log-convex. In the same vein, we extend a recent result of
Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan
polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising
from the Lambert $W$ function, we obtain a neat proof of the unimodality of the
latter sequence, which was proved previously by Kalugin and Jeffrey.
