  In this paper we study some extremal problems for the family
$S_g^0(\mathbb{B}_X)$ of normalized univalent mappings with $g$-parametric
representation on the unit ball $\mathbb{B}_X$ of an $n$-dimensional
JB$^*$-triple $X$ with $r\geq 2$, where $r$ is the rank of $X$ and $g$ is a
convex (univalent) function on the unit disc $\mathbb{U}$, which satisfies some
natural assumptions. We obtain sharp coefficient bounds for the family
$S_g^0(\mathbb{B}_X)$, and examples of bounded support points for various
subsets of $S_g^0(\mathbb{B}_X)$. Our results are generalizations to bounded
symmetric domains of known recent results related to support points for
families of univalent mappings on the Euclidean unit ball $\mathbb{B}^n$ and
the unit polydisc $\mathbb{U}^n$ in $\mathbb{C}^n$. Certain questions will be
also mentioned. Finally, we point out sharp coefficient bounds and bounded
support points for the family $S_g^0(\mathbb{B}^n)$ and for special compact
subsets of $S_g^0(\mathbb{B}^n)$, in the case $n\geq 2$.
