Support Points and Extreme Points for Mappings with \(A\) -Parametric Represen

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• 作者：Ian Graham ; Hidetaka Hamada ; Gabriela Kohr…
• 关键词：Subordination chains ; Parametric representation ; Support point ; Extreme point
• 刊名：Journal of Geometric Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：26
• 期：2
• 页码：1560-1595
• 全文大小：712 KB
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• 作者单位：Ian Graham (1)
  Hidetaka Hamada (2)
  Gabriela Kohr (3)
  Mirela Kohr (3)
  
  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada
  2. Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku, Fukuoka, 813-8503, Japan
  3. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Differential Geometry
  Convex and Discrete Geometry
  Fourier Analysis
  Abstract Harmonic Analysis
  Dynamical Systems and Ergodic Theory
  Global Analysis and Analysis on Manifolds
  
• 出版者：Springer New York
• ISSN：1559-002X

文摘

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 \(\tilde{\fancyscript{R}}_{\log M}(\mathrm{id}_{\mathbb {B}^2},{\fancyscript{N}}_A)\).