Support Points and Extreme Points for Mappings with $A$-Parametric Representation in $\mathbb {C}^{n}$

Abstract

Let $A\in L({\mathbb {C}}^n)$ be a linear operator such that $k_+(A)<2m(A)$, where $k_+(A)$ is the upper exponential index of $A$ and $m(A)=\min \{\mathfrak {R}\langle A(z),z\rangle :\Vert z\Vert =1\}$. In this paper we are concerned with variations of $A$-normalized univalent subordination chains on the Euclidean unit ball $\mathbb {B}^n$ in $\mathbb {C}^n$. We also obtain growth results for the generating vector fields and the transition mappings associated with $A$-normalized univalent subordination chains $f(z,t)$ that satisfy some regularity assumptions. Further, we give examples of normalized biholomorphic mappings on $\mathbb {B}^n$ which are not extreme/support points for the compact family $S_A^0(\mathbb {B}^n)$ of mappings with $A$-parametric representation on $\mathbb {B}^n$. Next, we consider the (normalized) time-$\log M$-reachable family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^n},{\fancyscript{N}}_A)$ generated by the Carathéodory mappings, where $M\in (1,\infty )$ and $k_+(A)<2m(A)$. We prove a result related to the support points $f$ of the family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^n},{\fancyscript{N}}_A)$ and the associated $A$-univalent subordination chains $f(z,t)$ such that $f=f(\cdot ,0)$. In particular, if $f\in \tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^n},{\fancyscript{N}}_A)$, then $f\not \in (\mathrm{supp}\, S_A^0(\mathbb {B}^n)\cup \mathrm{ex}\, S_A^0(\mathbb {B}^n))$. In the last part of the paper, using some ideas based on a shearing process that has been introduced recently by F. Bracci, we obtain a sharp estimate for $\Big |\frac{\partial ^2 f_1}{\partial z_2^2}(0)\Big |$, when $f=(f_1,f_2)\in S_A^0(\mathbb {B}^2)$ and $A$ is a diagonal matrix whose diagonal elements are $\lambda$ and $1, \lambda \in [1,2)$. This result allows us to construct an example of a bounded support mapping for the family $S_A^0(\mathbb {B}^2)$, but which does not belong to any reachable family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)$, for all $M>1$. This example provides a basic difference between the theory of bounded univalent mappings on the unit disc $U$ and that on the unit ball $\mathbb {B}^n$ in $\mathbb {C}^n, n\ge 2$. Finally, we construct an example of a support point $F_A^M$ for the reachable family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)$, but which is not a support point/extreme point for the family $S_A^0(\mathbb {B}^2)$, and we conjecture that the mapping $F_A^M$ is also an extreme point for the family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)$.