Skip to main content
SpringerLink
Log in

The Roper-Suffridge extension operator and classes of biholomorphic mappings

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper we give a survey about the Roper-Suffridge extension operator and the developments in the theory of univalent mappings in several variables to which it has led. We begin with the basic geometric properties (most of which now have a number of different proofs) and discuss relations with the theory of Loewner chains and generalizations and modifications of the operator, some of which are very recent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Roper K, Suffridge T J. Convex mappings on the unit ball of ℂn, J Anal Math, 1995, 65: 333–347

    MathSciNet  MATH  Google Scholar 

  2. Suffridge T J. Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes in Math, New York: Springer-Verlag, 1976, 599: 146–159

    Google Scholar 

  3. Suffridge T J. Holomorphic mappings of domains in ℂn onto convex domains. In: Geometric Function Theory in Several Complex Variables. River Edge, NJ: World Sci Publ, 2004: 295–309

    Google Scholar 

  4. Gong S. Convex and Starlike Mappings in Several Complex Variables. Dordrecht: Kluwer Acad Publ, 1998

    MATH  Google Scholar 

  5. Muir J R, Suffridge T J. Unbounded convex mappings of the ball in ℂn. Proc Amer Math Soc, 2001; 129: 3389–3393

    Article  MathSciNet  MATH  Google Scholar 

  6. Muir J R, Suffridge T J. Construction of convex mappings of p-balls in ℂ2. Comput Methods Funct Theory, 2004, 4: 21–34

    MathSciNet  MATH  Google Scholar 

  7. Muir J R, Suffridge T J. A generalization of half-plane mappings to the ball in ℂn. Trans Amer Math Soc, to appear

  8. Muir J R, Suffridge T J. Extreme points for convex mappings of B n. J Anal Math, to appear

  9. Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions, New York: Marcel Dekker Inc, 2003

    MATH  Google Scholar 

  10. Graham I, Hamada H, Kohr G, Suffridge T. Extension operators for locally univalent mappings. Michigan Math J, 2002, 50: 37–55

    Article  MathSciNet  MATH  Google Scholar 

  11. Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J Anal Math, 2000, 81: 331–342

    MathSciNet  MATH  Google Scholar 

  12. Gong S, Liu T S. On the Roper-Suffridge extension operator. J Anal Math, 2002, 88: 397–404

    MathSciNet  MATH  Google Scholar 

  13. Hamada H, Kohr G, Kohr M. Parametric representation and extension operators for biholomorphic mappings on some Reinhardt domains. Complex Variables, 2005, 50: 507–519

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu X. The generalized Roper-Suffridge extension operator for some biholomorphic mappings. J Math Anal Appl, to appear

  15. Graham I, Kohr G, Kohr M. Loewner chains and the Roper-Suffridge extension operator. J Math Anal Appl, 2000, 247: 448–465

    Article  MathSciNet  MATH  Google Scholar 

  16. Hamada H, Kohr G. Subordination chains and the growth theorem of spirallike mappings. Mathematica (Cluj), 2000, 42(65): 155–163

    MathSciNet  Google Scholar 

  17. Graham I. Growth and covering theorems associated with the Roper-Suffridge extension operator. Proc Amer Math Soc, 1999, 127: 3215–3220

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu M S, Zhu Y. On the generalized Roper-Suffridge extension operator in Banach spaces. Int J Math Math Sci, 2005, 8: 1171–1187

    Article  MathSciNet  Google Scholar 

  19. Liu X, Liu T. The generalized Roper-Suffridge extension operator for locally biholomorphic mappings. Chin Quart J Math, 2003, 18: 221–229

    MATH  Google Scholar 

  20. Zhu Y, Liu M S. The generalized Roper-Suffridge extension operator in Banach spaces, II. J Math Anal Appl, 2005, 303: 530–544

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu T. The growth theorems and covering theorems for biholomorphic mappings on classical domains, Doctoral Thesis, Univ Sci Tech China, 1989

  22. Graham I, Varolin D. Bloch constants in one and several variables. Pacific J Math, 1996, 174: 347–357

    MathSciNet  MATH  Google Scholar 

  23. Liczberski P, Starkov V. Distortion theorems for biholomorphic convex mappings in ℂn. J Math Anal Appl, 2002, 274: 495–504

    Article  MathSciNet  MATH  Google Scholar 

  24. Hamada H, Kohr G. Roper-Suffridge extension operator and the lower bound for the distortion. J Math Anal Appl, 2004, 300: 143–151

    Article  MathSciNet  Google Scholar 

  25. Gong S, Liu T. Distortion theorems for biholomorphic convex mappings on bounded convex circular domains. In: Proc of the Fifth International Colloquium on Complex Analysis, 1997, 73–80; Chin Ann Math Ser B, 1999, 20: 297–304

  26. Pfaltzgraff J A, Suffridge T J. Norm order and geometric properties of holomorphic mappings in ℂn. J Anal Math, 2000, 82: 285–313

    MathSciNet  MATH  Google Scholar 

  27. Liczberski P, Starkov V. On two conjectures for convex biholomorphic mappings in ℂn. J Anal Math, 2004, 94: 377–383

    Article  MathSciNet  MATH  Google Scholar 

  28. Pfaltzgraff J A. Subordination chains and univalence of holomorphic mappings in ℂn. Math Ann, 1974, 210: 55–68

    Article  MathSciNet  MATH  Google Scholar 

  29. Graham I, Hamada H, Kohr G. Parametric representation of univalent mappings in several complex variables. Canadian J Math, 2002, 54: 324–351

    MathSciNet  MATH  Google Scholar 

  30. Graham I, Kohr G, Kohr G. Loewner chains and parametric representation in several complex variables. J Math Anal Appl, 2003, 281: 425–438

    Article  MathSciNet  MATH  Google Scholar 

  31. Graham I, Kohr G, Kohr M. Basic properties of Loewner chains in several complex variables. In: FitzGerald C H, Gong S, eds. Geometric Function Theory in Several Complex Variables, World Sci Publ, 2004: 165–181

  32. Poreda T. On the univalent subordination chains of holomorphic mappings in Banach spaces. Commentationes Math, 1989, 128: 295–304

    MathSciNet  Google Scholar 

  33. Poreda T. On the univalent holomorphic maps of the unit polydisc in ℂn which have the parametric representation, I — the geometrical properties. Ann Univ Mariae Curie Sklodowska, Sect A, 1987, 41: 105–113

    MathSciNet  MATH  Google Scholar 

  34. Poreda T. On the univalent holomorphic maps of the unit polydisc in ℂn which have the parametric representation, II — the necessary conditions and the sufficient conditions. Ann Univ Mariae Curie Sklodowska, Sect A, 1987, 41: 115–121

    MathSciNet  MATH  Google Scholar 

  35. Graham I, Kohr G, Pfaltzgraff J A. Parametric representation and linear functionals associated with extension operators for biholomorphic mappings. Rev Roum Math Pures Appl, to appear

  36. Gong S, Liu T S. The generalized Roper-Suffridge extension operator. J Math Anal Appl, 2003, 284: 425–434

    Article  MathSciNet  MATH  Google Scholar 

  37. Muir J R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5: 237–251

    MathSciNet  MATH  Google Scholar 

  38. Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator, Mathematica (Cluj), 2006, 48(71): 41–48

    MathSciNet  Google Scholar 

  39. Liu T, Xu Q. Loewner chains associated with the generalized Roper-Suffridge extension operator. J Math Anal Appl, 2006, 322: 107–120

    Article  MathSciNet  MATH  Google Scholar 

  40. Pfaltzgraff J A, Suffridge T J. An extension theorem and linear invariant families generated by starlike maps. Ann Univ Mariae Curie Sklodowska, Sect A, 1999, 53: 193–207

    MathSciNet  MATH  Google Scholar 

  41. Pommerenke C. Linear-invariante Familien analytischer Funktionen. I Math Ann, 1964, 155: 108–154

    Article  MathSciNet  MATH  Google Scholar 

  42. Barnard R W, FitzGerald C H, Gong S. A distortion theorem for biholomorphic mappings in ℂ2. Trans Amer Math Soc, 1994, 344: 907–924

    Article  MathSciNet  MATH  Google Scholar 

  43. Pfaltzgraff J A. Distortion of locally biholomorphic maps of the n-ball. Complex Variables, 1997, 33: 239–253

    MathSciNet  MATH  Google Scholar 

  44. Pfaltzgraff J A, Suffridge T J. Linear invariance, order and convex maps in ℂn. Complex Variables, 1999, 40: 35–50

    MathSciNet  MATH  Google Scholar 

  45. Godula J, Liczberski P, Starkov V. Order of linearly invariant family of mappings in ℂn. Complex Variables, 2000, 42: 89–96

    MathSciNet  MATH  Google Scholar 

  46. Liczberski P, Starkov V. Linearly invariant families of holomorphic mappings of a ball, The dimension reduction method. Siberian Math J, 2001, 42: 715–730

    Article  MathSciNet  Google Scholar 

  47. Pfaltzgraff J A. Distortion of locally biholomorphic maps of the n-ball. Erratum. Complex Variables, 2001, 45: 197–200

    MathSciNet  Google Scholar 

  48. Roper K, Suffridge T J. Convexity properties of holomorphic mappings in ℂn. Trans Amer Math Soc, 1999, 351: 1803–1833

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Graham Ian

  2. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Kohr Gabriela

Authors
  1. Graham Ian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kohr Gabriela
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Graham, I., Kohr, G. The Roper-Suffridge extension operator and classes of biholomorphic mappings. SCI CHINA SER A 49, 1539–1552 (2006). https://doi.org/10.1007/s11425-006-2051-z

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-006-2051-z

Keywords

MSC(2000)

Search

Navigation