Support points for families of univalent mappings on bounded symmetric domains

Abstract

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${}^{\ast }$-triple $X$ with $r\ge 2$, 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 $n\ge 2$.

SCIENCE CHINA Mathematics