Spirallike mappings and univalent subordination chains in ℂ n

Ian Graham; Hidetaka Hamada; Gabriela Kohr; Mirela Kohr

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2008)

  • Volume: 7, Issue: 4, page 717-740
  • ISSN: 0391-173X

Abstract

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In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in ℂ n . To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent subordination chains. We prove that such a spirallike mapping can be imbedded as the first element of a univalent subordination chain, and we present various particular cases and examples. If the matrix-valued mapping is constant, we obtain the usual notion of spirallikeness with respect to a linear operator.

How to cite

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Graham, Ian, et al. "Spirallike mappings and univalent subordination chains in $\mathbb {C}^n$." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 7.4 (2008): 717-740. <http://eudml.org/doc/272271>.

@article{Graham2008,
abstract = {In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in $\mathbb \{C\}^n$. To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent subordination chains. We prove that such a spirallike mapping can be imbedded as the first element of a univalent subordination chain, and we present various particular cases and examples. If the matrix-valued mapping is constant, we obtain the usual notion of spirallikeness with respect to a linear operator.},
author = {Graham, Ian, Hamada, Hidetaka, Kohr, Gabriela, Kohr, Mirela},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {4},
pages = {717-740},
publisher = {Scuola Normale Superiore, Pisa},
title = {Spirallike mappings and univalent subordination chains in $\mathbb \{C\}^n$},
url = {http://eudml.org/doc/272271},
volume = {7},
year = {2008},
}

TY - JOUR
AU - Graham, Ian
AU - Hamada, Hidetaka
AU - Kohr, Gabriela
AU - Kohr, Mirela
TI - Spirallike mappings and univalent subordination chains in $\mathbb {C}^n$
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2008
PB - Scuola Normale Superiore, Pisa
VL - 7
IS - 4
SP - 717
EP - 740
AB - In this paper we consider non-normalized univalent subordination chains and the connection with the Loewner differential equation on the unit ball in $\mathbb {C}^n$. To this end, we study the most general form of the initial value problem for the transition mapping, and prove the existence and uniqueness of solutions. Also we introduce the notion of generalized spirallikeness with respect to a measurable matrix-valued mapping, and investigate this notion from the point of view of non-normalized univalent subordination chains. We prove that such a spirallike mapping can be imbedded as the first element of a univalent subordination chain, and we present various particular cases and examples. If the matrix-valued mapping is constant, we obtain the usual notion of spirallikeness with respect to a linear operator.
LA - eng
UR - http://eudml.org/doc/272271
ER -

References

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  1. [1] F. F. Bonsall and J. Duncan, “Numerical Ranges. II”, Cambridge Univ. Press, 1973. Zbl0262.47001MR442682
  2. [2] E. A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations”, McGraw-Hill Book Co., New York-Toronto-London, 1955. Zbl0064.33002MR69338
  3. [3] Yu. L. Daleckii and M.G. Krein, “Stability of Solutions of Differential Equations in a Banach Space”, Translations of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, R.I., 1974. Zbl0286.34094MR352639
  4. [4] N. Dunford and J.T. Schwartz, “Linear Operators. I”, Interscience Publ., Inc., New York, 1966. Zbl0084.10402
  5. [5] M. Elin, S. Reich and D. Shoikhet, Complex dynamical systems and the geometry of domains in Banach spaces, Dissertationes Math.427 (2004), 1–62. Zbl1060.37038MR2071666
  6. [6] I. Graham, H. Hamada and G. Kohr, Parametric representation of univalent mappings in several complex variables, Canad. J. Math.54 (2002), 324–351. Zbl1004.32007MR1892999
  7. [7] I. Graham, H. Hamada, G. Kohr and M. Kohr, Parametric representation and asymptotic starlikeness in ℂ n , Proc. Amer. Math. Soc.136 (2008), 267–302. Zbl1157.32016MR2425737
  8. [8] I. Graham, H. Hamada, G. Kohr and M. Kohr, Asymptotically spirallike mappings in several complex variables, J. Anal. Math.105 (2008), 267–302. Zbl1148.32009MR2438427
  9. [9] I. Graham and G. Kohr, “Geometric Function Theory in One and Higher Dimensions”, Marcel Dekker Inc., New York, 2003. Zbl1042.30001MR2017933
  10. [10] I. Graham, G. Kohr and M. Kohr, Loewner chains and the Roper-Suffridge extension operator, J. Math. Anal. Appl.247 (2000), 448–465. Zbl0965.32008MR1769088
  11. [11] I. Graham, G. Kohr and M. Kohr, Loewner chains and parametric representation in several complex variables, J. Math. Anal. Appl.281 (2003), 425–438. Zbl1029.32004MR1982664
  12. [12] K. Gurganus, Φ -like holomorphic functions in ℂ n and Banach spaces, Trans. Amer. Math. Soc.205 (1975), 389–406. Zbl0299.32018MR374470
  13. [13] K. E. Gustafson and D.K.M. Rao, “Numerical Range. The Field of Values of Linear Operators and Matrices”, Springer-Verlag, New York, 1997. Zbl0874.47003MR1417493
  14. [14] H. Hamada and G. Kohr, Subordination chains and the growth theorem of spirallike mappings, Mathematica (Cluj) 42 (65) (2000), 153–161. Zbl1027.46094MR1988620
  15. [15] H. Hamada and G. Kohr, An estimate of the growth of spirallike mappings relatve to a diagonal matrix, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 55 (2001), 53–59. Zbl1018.32007MR1845250
  16. [16] L. Harris, The numerical range of holomorphic functions in Banach spaces, Amer. J. Math.93 (1971), 1005–1019. Zbl0237.58010MR301505
  17. [17] L. A. Harris, S. Reich and D. Shoikhet, Dissipative holomorphic functions, Bloch radii, and the Schwarz lemma, J. Anal. Math. 82 (2000), 221–232. Zbl0972.46029MR1799664
  18. [18] G. Kohr, Using the method of Löwner chains to introduce some subclasses of biholomorphic mappings in ℂ n , Rev. Roumaine Math. Pures Appl.46 (2001), 743–760. Zbl1036.32014MR1929522
  19. [19] J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings in ℂ n , Math. Ann.210 (1974), 55–68. Zbl0275.32012MR352510
  20. [20] J. A. Pfaltzgraff and T.J. Suffridge, An extension theorem and linear invariant families generated by starlike maps, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 53 (1999), 193–207. Zbl0996.32006MR1778828
  21. [21] C. Pommerenke, Über die subordination analytischer funktinonen, J. Reine Angew. Math.218 (1965), 159–173. Zbl0184.30601MR180669
  22. [22] C. Pommerenke, “Univalent functions”, Vandenhoeck & Ruprecht, Göttingen, 1975. Zbl0298.30014
  23. [23] T. Poreda, On the univalent holomorphic maps of the unit polydisc in ℂ n which have the parametric representation, I-the geometrical properties, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 41 (1987), 105–113. Zbl0698.32004MR1049182
  24. [24] T. Poreda, On the univalent holomorphic maps of the unit polydisc in ℂ n which have the parametric representation, II-the necessary conditions and the sufficient conditions, Ann. Univ. Mariae Curie-Skłodowska, Sect. A. 41 (1987), 115–121. Zbl0698.32005MR1049183
  25. [25] T. Poreda, On the univalent subordination chains of holomorphic mappings in Banach spaces, Comment. Math. Prace Mat.28 (1989), 295–304. Zbl0694.46033MR1024945
  26. [26] T. Poreda, On generalized differential equations in Banach spaces, Dissertationes Math.310 (1991), 1–50. Zbl0745.35048MR1104523
  27. [27] S. Reich and D. Shoikhet, “Nonlinear Semigroups, Fixed Points, and Geometry of Domains in Banach Spaces”, Imperial College Press, London, 2005. Zbl1089.46002MR2022955
  28. [28] K. Roper and T. J. Suffridge, Convex mappings on the unit ball of ℂ n , J. Anal. Math.65 (1995), 333–347. Zbl0846.32006MR1335379
  29. [29] T. J. Suffridge, Starlike and convex maps in Banach spaces, Pacific J. Math.46 (1973), 575–589. Zbl0263.30016MR374914
  30. [30] T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, In: “Lecture Notes in Math.”, Springer-Verlag 599 (1977), 146–159. Zbl0356.32004MR450601
  31. [31] K. Yosida, “Functional Analysis”, Springer-Verlag, 1965. 

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