Skip to main content
SpringerLink
Log in

A proof of a weak version of the Bieberbach conjecture in several complex variables

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, the sharp estimates of all homogeneous expansions for a subclass of starlike mappings on the unit ball in complex Banach spaces are first established. Meanwhile, the sharp estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in ℂn are also obtained. Our results show that a weak version of the Bieberbach conjecture in several complex variables is proved, and the obtained conclusions reduce to the classical results in one complex variable.

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. de Branges L. A proof of the Bieberbach conjecture. Acta Math, 1985, 154: 137–152

    Article  MATH  MathSciNet  Google Scholar 

  2. Cartan H. Sur la possibilité d’ étendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalentes, Note added to Montel P, Lecons sur les Fonctions Univalentes ou Mutlivalentes. Paris: Gauthier-Villars, 1933

    Google Scholar 

  3. Gong S. The Bieberbach Conjecture. Providence, RI: Amer Math Soc, 1999

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  5. Hamada H, Honda T. Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin Ann Math Ser B, 2008, 29: 353–368

    Article  MATH  MathSciNet  Google Scholar 

  6. Honda T. The growth theorem for k-fold symmetric convex mappings. Bull London Math Soc, 2002, 34: 717–724

    Article  MATH  MathSciNet  Google Scholar 

  7. Lin Y Y, Hong Y. Some properties of holomorphic maps in Banach spaces (in Chinese). Acta Math Sin, 1995, 38: 234–241

    MATH  MathSciNet  Google Scholar 

  8. Liu X S. On the quasi-convex mappings on the unit polydisk in ℂn. J Math Anal Appl, 2007, 335: 43–55

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu X S, Liu T S. The sharp estimate of the third homogeneous expansion for a class of starlike mappings of order α on the unit polydisk in ℂn. Acta Math Sci Ser B, 2012, 32: 752–764

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Suffridge T J. Starlike and convex maps in Banach spaces. Pacif Math J, 1973, 46: 575–589

    Article  MATH  MathSciNet  Google Scholar 

  12. Suffridge T J. The principle of subordination applied to functions of several variables. Pacif Math J, 1970, 33: 241–248

    Article  MATH  MathSciNet  Google Scholar 

  13. Xu Q H, Liu T S. On coefficient estimates for a class of holomorphic mappings. Sci Chin Ser A, 2009, 52: 677–686

    Article  MATH  Google Scholar 

  14. Zhang W J, Dong D Z, Wang Y Z. The growth theorem for convex maps on the Banach space (in Chinese). Chin Quartly J Math, 1992 7: 84–87

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computation Science, Lingnan Normal University, Zhanjiang, 524048, China

    XiaoSong Liu

  2. Department of Mathematics, Huzhou Teachers College, Huzhou, 313000, China

    TaiShun Liu

  3. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330027, China

    QingHua Xu

Authors
  1. XiaoSong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. TaiShun Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. QingHua Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XiaoSong Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Liu, T. & Xu, Q. A proof of a weak version of the Bieberbach conjecture in several complex variables. Sci. China Math. 58, 2531–2540 (2015). https://doi.org/10.1007/s11425-015-5016-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-015-5016-2

Keywords

MSC(2010)

Search

Navigation