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