Extremal properties associated with univalent subordination chains in \(\mathbb {C}^n\)
详细信息 查看全文
- 作者：Ian Graham (1)
Hidetaka Hamada (2)
Gabriela Kohr (3)
Mirela Kohr (3) - 关键词：32H02 ; 30C45
- 刊名：Mathematische Annalen
- 出版年：2014
- 出版时间：June 2014
- 年：2014
- 卷：359
- 期：1-2
- 页码：61-99
- 全文大小：
- 参考文献：1. Abate, M., Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: The evolution of Loewner’s differential equations. Eur. Math. Soc. Newsl. 78, 31-8 (2010)
2. Arosio, L.: Resonances in Loewner equations. Adv. Math. 227, 1413-435 (2011) CrossRef
3. Arosio, L., Bracci, F.: Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds. Anal. Math. Phys. 1, 337-50 (2011) CrossRef
4. Arosio, L., Bracci, F., Hamada, H., Kohr, G.: An abstract approach to Loewner chains. J. Anal. Math. 119, 89-14 (2013) CrossRef
5. Arosio, L., Bracci, F., Wold, F.E.: Solving the Loewner PDE in complete hyperbolic starlike domains of \(\mathbb{C}^n\) . Adv. Math. 242, 209-16 (2013) CrossRef
6. Becker, J.: über die L?sungsstruktur einer Differentialgleichung in der konformen Abbildung. J. Reine Angew. Math. 285, 66-4 (1976)
7. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation II: complex hyperbolic manifolds. Math. Ann. 344, 947-62 (2009) CrossRef
8. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Semigroups versus evolution families in the Loewner theory. J. Anal. Math. 115, 273-92 (2011) CrossRef
9. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution families and the Loewner equation I: the unit disk. J. Reine Angew. Math. 672, 1-7 (2012)
10. Brickman, L., Wilken, D.R.: Support points of the set of univalent functions. Proc. Am. Math. Soc. 42, 523-28 (1974) CrossRef
11. Cartan, H.: Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes, 129-55. Le?ons sur les fonctions univalentes ou multivalentes, Gauthier-Villars, Paris, Note added to P. Montel (1933)
12. Daleckii, Yu.L., Krein, M.G.: Stability of solutions of differential equations in Banach space. Transl. Math. Monogr., vol. 43. American Mathematical Society, Providence (1974)
13. Duren, P.: Univalent Functions. Springer, New York (1983)
14. Duren, P., Graham, I., Hamada, H., Kohr, G.: Solutions for the generalized Loewner differential equation in several complex variables. Math. Ann. 347, 411-35 (2010) CrossRef
15. Elin, M.: Extension operators via semigroups. J. Math. Anal. Appl. 377, 239-50 (2011) CrossRef
16. Elin, M., Reich, S., Shoikhet, D.: Complex dynamical systems and the geometry of domains in Banach spaces. Diss. Math. 427, 1-2 (2004)
17. Friedland, S., Schiffer, M.: On coefficient regions of univalent functions. J. Anal. Math. 31, 125-68 (1977) CrossRef
18. Goodman, G.S.: Univalent Functions and Optimal Control, Ph.D. Thesis, Stanford Univ. (1968)
19. Gong, S.: Convex and Starlike Mappings in Several Complex Variables. Kluwer, Dordrecht (1998) CrossRef
20. Graham, I., Hamada, H., Kohr, G.: Parametric representation of univalent mappings in several complex variables. Can. J. Math. 54, 324-51 (2002) CrossRef
21. Graham, I., Hamada, H., Kohr, G.: Extension operators and subordination chains. J. Math. Anal. Appl. 386, 278-89 (2012) CrossRef
22. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically spirallike mappings in several complex variables. J. Anal. Math. 105, 267-02 (2008) CrossRef
23. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Parametric representation and asymptotic starlikeness in \(\mathbb{C}^n\) . Proc. Am. Math. Soc. 136, 3963-973 (2008) CrossRef
24. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Spirallike mappings and univalent subordination chains in \(\mathbb{C}^n\) . Ann. Scuola Norm. Sup. Pisa-Cl. Sci. 7, 717-40 (2008)
25. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Extreme points, support points and the Loewner variation in several complex variables. Sci. China Math. 55, 1353-366 (2012) CrossRef
26. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Univalent subordination chains in reflexive complex Banach spaces. Contemp. Math. (AMS) 591, 83-11 (2013) CrossRef
27. Graham, I., Hamada, H., Kohr, G., Kohr, M.: Asymptotically spirallike mappings in reflexive complex Banach spaces. Complex Anal. Oper. Theory 7, 1909-927 (2013) CrossRef
28. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker Inc, New York (2003)
29. Graham, I., Kohr, G., Kohr, M.: Loewner chains and parametric representation in several complex variables. J. Math. Anal. Appl. 281, 425-38 (2003) CrossRef
30. Graham, I., Kohr, G., Pfaltzgraff, J.A.: Parametric representation and linear functionals associated with extension operators for biholomorphic mappings. Rev. Roumaine Math. Pures Appl. 52, 47-8 (2007)
31. Gurganus, K.: \(\Phi \) -like holomorphic functions in \(\mathbb{C}^n\) and Banach spaces. Trans. Am. Math. Soc. 205, 389-06 (1975)
32. Hallenbeck, D.J., MacGregor, T.H.: Linear Problems and Convexity Techniques in Geometric Function Theory. Pitman, Boston (1984)
33. Hamada, H.: Polynomially bounded solutions to the Loewner differential equation in several complex variables. J. Math. Anal. Appl. 381, 179-86 (2011) CrossRef
34. Hamada, H., Kohr, G., Muir Jr, J.R.: Extensions of \(L^d\) -Loewner chains to higher dimensions. J. Anal. Appl. 120, 357-92 (2013)
35. Harris, L.: The numerical range of holomorphic functions in Banach spaces. Am. J. Math. 93, 1005-019 (1971) CrossRef
36. Jurdjevic, V.: Geometric Control Theory. Cambridge Univ Press, Cambridge (1997)
37. Kirwan, W.E.: Extremal properties of slit conformal mappings. In: Brannan, D., Clunie, J. (eds.) Aspects of Contemporary Complex Analysis, pp. 439-49. Academic Press, London (1980)
38. Kirwan, W.E., Schober, G.: New inequalities from old ones. Math. Z. 180, 19-0 (1982) CrossRef
39. Lee, E., Markus, L.: Foundations of Optimal Control Theory. Wiley, New York (1967)
40. Loewner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I. Math. Ann. 89, 103-21 (1923) CrossRef
41. MacGregor, T.H., Wilken, D.R.: Extreme points and support points. In: Kühnau, R. (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. I, pp. 371-92. Elsevier, Amsterdam (2002)
42. Muir, J.R.: A class of Loewner chain preserving extension operators. J. Math. Anal. Appl. 337, 862-79 (2008) CrossRef
43. Pell, R.: Support point functions and the Loewner variation. Pacific J. Math. 86, 561-64 (1980) CrossRef
44. Pfaltzgraff, J.A.: Subordination chains and univalence of holomorphic mappings in \(\mathbb{C}^n\) . Math. Ann. 210, 55-8 (1974) CrossRef
45. Pfaltzgraff, J.A.: Subordination chains and quasiconformal extension of holomorphic maps in \(\mathbb{C}^n\) . Ann. Acad. Scie. Fenn. Ser. A I Math. 1, 13-5 (1975)
46. Pfluger, A.: Lineare extremal probleme bei schlichten Funktionen. Ann. Acad. Sci. Fenn. Ser. A I., 489 (1971)
47. Pommerenke, C.: über die subordination analytischer funktionen. J. Reine Angew. Math. 218, 159-73 (1965)
48. Pommerenke, C.: Univalent Functions. Vandenhoeck & Ruprecht, G?ttingen (1975)
49. Poreda, T.: On Generalized Differential Equations in Banach Spaces. Diss. Math. 310, 1-0 (1991)
50. Poreda, T.: On the univalent holomorphic maps of the unit polydisc in \(\mathbb{C}^n\) which have the parametric representation, I-the geometrical properties. Ann. Univ. Mariae Curie Skl. Sect. A. 41, 105-13 (1987)
51. Poreda, T.: On the univalent holomorphic maps of the unit polydisc in \(\mathbb{C}^n\) which have the parametric representation, II-the necessary conditions and the sufficient conditions. Ann. Univ. Mariae Curie Skl. Sect. A. 41, 115-21 (1987)
52. Prokhorov, D.V.: Bounded univalent functions. In: Kühnau, R. (ed.) Handbook of Complex Analysis: Geometric Function Theory, vol. I, pp. 207-28, Elsevier, Amsterdam (2002)
53. Range, M.: Holomorphic Functions and Integral Representations in Several Complex Variables. Springer, New York (1986) CrossRef
54. Reich, S., Shoikhet, D.: Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces. Imperial College Press, London (2005) CrossRef
55. Roper, K., Suffridge, T.J.: Convex mappings on the unit ball of \(\mathbb{C}^n\) . J. Anal. Math. 65, 333-47 (1995) CrossRef
56. Roth, O.: Control theory in \({\cal H}(\mathbb{D})\) . Diss. Bayerischen Univ, Wuerzburg (1998)
57. Roth, O.: A remark on the Loewner differential equation, Comput. Methods and Function Theory 1997 (Nicosia), Ser. Approx. Decompos, vol. 11, pp. 461-69. World Sci. Publ. River Edge (1999)
58. Schleissinger, S.: On support points of the class \(S^0(B^n)\) . Proc. Am. Math. Soc. (2013, to appear)
59. Suffridge, T.J.: Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions. In: Lecture Notes Math., vol. 599, pp. 146-59. Springer, Berlin (1977)
60. Voda, M.: Solution of a Loewner chain equation in several complex variables. J. Math. Anal. Appl. 375, 58-4 (2011) CrossRef
61. Voda, M.: Loewner theory in several complex variables and related problems, Ph.D thesis, Univ. Toronto (2011) - 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, Babe?-Bolyai University, 1 M. Kog?lniceanu Str., 400084?, Cluj-Napoca, Romania - ISSN：1432-1807
文摘
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.