Shearing Process and an Example of a Bounded Support Function in \(S^0(\mathbb B^2)\)

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• 作者：Filippo Bracci (1)
  
  1. Dipartimento di Matematica ; Universit脿 di Roma 鈥淭or Vergata鈥? Via Della Ricerca Scientifica 1 ; 00133 ; Rome ; Italy
  
• 关键词：Support points ; Shears ; Geometric function theory ; Loewner theory ; Univalent maps ; Primary 30C50 ; Secondary 32A70 ; 32A10
• 刊名：Computational Methods and Function Theory
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：15
• 期：1
• 页码：151-157
• 全文大小：197 KB
• 参考文献：1. Bracci, F., Graham, I., Hamada, H., Kohr, G.: Variation of Loewner chains, extreme and support points in the class \(S^0\) in higher dimensions. Preprint (2014)
  2. Graham, I, Kohr, G (2003) Geometric function theory in one and higher dimensions. Marcel Dekker Inc., New York
  3. Graham, I, Hamada, H, Kohr, G, Kohr, M (2008) Parametric representation and asymptotic starlikeness in $$\mathbb{C}^{n}$$ C n. Proc. Am. Math. Soc. 136: pp. 3963-3973 CrossRef
  4. Graham, I, Hamada, H, Kohr, G, Kohr, M (2012) Extreme points, support points, and the Loewner variation in several complex variables. Sci. China Math. 55: pp. 1353-1366 CrossRef
  5. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Extremal properties associated with univalent subordination chains in \(\mathbb{C}^{n}\) . Math. Ann. 359(1鈥?), 61鈥?9 (2014)
  6. Graham, I, Hamada, H, Kohr, G (2012) Extension operators and subordination chains.. J. Math. Anal. Appl. 386: pp. 278-289 CrossRef
  7. Graham, I., Hamada, H., Kohr, G.: Extremal problems and \(g\) -Loewner chains in \(\mathbb{C}^{n}\) and reflexive complex Banach spaces. Preprint (2013)
  8. Muir, JR (2008) A class of Loewner chain preserving extension operators.. J. Math. Anal. Appl. 337: pp. 862-879 CrossRef
  9. Pfaltzgraff, JA, Suffridge, TJ (2007) Koebe invariant functions and extremal problems for holomorphic mappings in the unit ball of $$\mathbb{C}^{n}$$ C n. Comput. Methods Funct. Theory 7: pp. 379-399 CrossRef
  10. Pommerenke, Ch.: Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrb眉cher, Band XXV. Vandenhoeck and Ruprecht, G枚ttingen (1975)
  11. Roper, KA, Suffridge, TJ (1999) Convexity properties of holomorphic mappings in $$\mathbb{C}^{n}$$ C n. Trans. Amer. Math. Soc. 351: pp. 1803-1833 CrossRef
  12. Roth, O.: A variational method for the Loewner equation in higher dimensions. Preprint (2014)
  13. Schaeffer, A. C., Spencer, D. C.: Coefficient regions for Schlicht functions. In: Region of the Derivative of a Schlicht Function by Arthur Grad, vol. 35, pp. xv+311. American Mathematical Society Colloquium Publications, American Mathematical Society, New York (1950)
  14. Schleissinger, S.: On support points of the class \(S^0(B^n)\) . Proc. Am. Math. Soc. (2014, in press)
  15. Suffridge, T.J.: Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions. In: Lecture Notes in Mathematics, vol. 599, pp.146鈥?59. Springer, Berlin (1977)
  
• 刊物主题：Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable;
• 出版者：Springer Berlin Heidelberg
• ISSN：2195-3724

文摘

We introduce a process, that we call shearing, which for any given normal Loewner chain produces a normal Loewner chain made of shears automorphisms. As an application, and in stringent contrast to the one-dimensional case, we prove the existence of a starlike bounded function in the class \(S^0\) of the ball \(\mathbb B^2\) (in fact the restriction of a shear automorphism of \(\mathbb C^2\) ) which is a support point for a linear continuous functional.