Extremal properties associated with univalent subordination chains in $\mathbb {C}^n$

Abstract

For a linear operator $A\in L(\mathbb {C}^n)$, let $k_+(A)$ be the upper exponential index of $A$ and let $m(A)=\min \{\mathfrak {R}\langle A(z),z\rangle :\Vert z\Vert =1\}$. Under the assumption $k_+(A)<2m(A)$, we consider the family $S_A^0(B^n)$ of mappings which have $A$-parametric representation on the Euclidean unit ball $B^n$ in $\mathbb {C}^n$, i.e. $f\in S_A^0(B^n)$ if and only if there exists an $A$-normalized univalent subordination chain $f(z,t)$ such that $f=f(\cdot ,0)$ and $\{e^{-tA}f(\cdot ,t)\}_{t\ge 0}$ is a normal family on $B^n$. We prove that if $f=f(\cdot ,0)$ is an extreme point (respectively a support point) of $S_A^0(B^n)$, then $e^{-tA}f(\cdot ,t)$ is an extreme point of $S_A^0(B^n)$ for $t\ge 0$ (respectively a support point of $S_A^0(B^n)$ for $t\ge 0$). These results generalize to higher dimensions related results due to Pell and Kirwan. We also deduce an $n$-dimensional version of an extremal principle due to Kirwan and Schober. In the second part of the paper, we consider extremal problems related to bounded mappings in $S_A^0(B^n)$. To this end, we use ideas from control theory to investigate the (normalized) time-$\log M$-reachable family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{B^n},{\fancyscript{N}}_A)$ of (4.1) generated by the Carathéodory mappings, where $M\ge 1$ and $k_+(A)<2m(A)$. We prove that each mapping $f$ in the above reachable family can be imbedded as the first element of an $A$-normalized univalent subordination chain $f(z,t)$ such that $\{e^{-tA}f(\cdot ,t)\}_{t\ge 0}$ is a normal family and $f(\cdot ,\log M)=e^{A\log M}\mathrm{id}_{B^n}$. We also prove that the family $\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{B^n},{\fancyscript{N}}_A)$ is compact and we deduce a density result related to the same family, which involves the subset $\mathrm{ex}\,{\fancyscript{N}}_A$ of ${\fancyscript{N}}_A$ consisting of extreme points. These results are generalizations to $\mathbb {C}^n$ of related results due to Roth. Finally, we are concerned with extreme points and support points associated with compact families generated by extension operators.