Extremal Problems for Mappings with Generalized Parametric Representation in ${{\mathbb {C}}}^{n}$

Abstract

In this paper we are concerned with the family $\widetilde{S}^t_A(\mathbb {B}^n)$ ($t\ge 0$) of normalized biholomorphic mappings on the Euclidean unit ball $\mathbb {B}^n$ in ${\mathbb {C}}^n$ that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators $A\in \widetilde{\mathcal {A}}$, where $\widetilde{\mathcal {A}}$ is a family of measurable mappings from $[0,\infty )$ into $L({\mathbb {C}}^n)$ which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family $\widetilde{S}^t_A(\mathbb {B}^n)$, where $A\in \widetilde{\mathcal {A}}$. We prove that if $f(z,t)=V(t)^{-1}z+\cdots$ is a normal Loewner chain such that $V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)$ (resp. $V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)$), then $V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)$, for all $t\ge s$ (resp. $V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)$, for all $t\ge s$), where V(t) is the unique solution on $[0,\infty )$ of the initial value problem: $\frac{d V}{d t}(t)=-A(t)V(t)$, a.e. $t\ge 0$, $V(0)=I_n$. Also, we obtain an example of a bounded support point for the family $\widetilde{S}_A^t(\mathbb {B}^2)$, where $A\in \widetilde{\mathcal {A}}$ is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators $A\in \widetilde{\mathcal {A}}$, and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on $\mathbb {B}^n$. Useful examples and applications yield that the study of the family $\widetilde{S}^t_A(\mathbb {B}^n)$ for time-dependent operators $A\in \widetilde{\mathcal {A}}$ is basically different from that in the case of constant time-dependent linear operators.