Convex families of holomorphic mappings related to the convex mappings of the ball in $\mathbb {C}^n$

Abstract:

Unlike the case in one dimension, there is still much to learn about the basic nature of the family $\mathcal {K}(\mathbb {B})$ of normalized ($f(0)=0$, $Df(0)=I$, where $Df$ is the Fréchet derivative of $f$ and $I$ is the identity operator on $\mathbb {C}^n$) biholomorphic mappings $f$ of the Euclidean unit ball $\mathbb {B} \subseteq \mathbb {C}^n$ onto convex domains in $\mathbb {C}^n$ when $n\geq 2$. We consider its closed convex hull $\overline {\mathrm {co}} \mathcal {K}(\mathbb {B})$ in relation to the family $\mathcal {R}(\mathbb {B})$ of normalized holomorphic mappings $f\colon \mathbb {B} \rightarrow \mathbb {C}^n$ satisfying $\operatorname {Re} \langle f(z),z \rangle > \|z\|^2/2$ for $z\in \mathbb {B} \setminus \{0\}$, where $\langle \cdot ,\cdot \rangle$ and $\|\cdot \|$ are, respectively, the Hermitian inner product and Euclidean norm in $\mathbb {C}^n$. In dimension $n=1$, the sets are the same. Here, we identify some extreme points of $\overline {\mathrm {co}} \mathcal {K}(\mathbb {B})$ and use them to show that $\overline {\mathrm {co}} \mathcal {K}(\mathbb {B})$ is a proper subset of $\mathcal {R}(\mathbb {B})$ when $n\geq 2$. We also consider an extension operator related to $\mathcal {R}(\mathbb {B})$ that helps illustrate where the known extreme points of $\overline {\mathrm {co}} \mathcal {K}(\mathbb {B})$ lie in $\mathcal {R}(\mathbb {B})$ and make some observations on the related case of the unit polydisk.