Skip to main content
SpringerLink
Log in

On the Bombieri numbers for the class S

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

We derive a second variation formula of the Koebe function developed by Bombieri by using a linear version of the Loewner differential equation. We then express the formula in terms of classical orthogonal polynomials to consider the Bombieri numbers obtained by Greiner and Roth (Proc Am Math Soc 129(12):3657–3664, 2001) and by Prokhorov and Vasil’ev (Georgian Math J 12(4):743–761, 2005). We show a question raised by Bombieri on these numbers has a negative answer in the low order cases.

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

  • Abate, M., F. Bracci, M. Contreras, and S. Díaz-Madrigal. 2010. The evolution of Loewner’s differential equations. European Mathematical Society Newsletter 78: 31–38.

    MathSciNet  MATH  Google Scholar 

  • Askey, R. 1975. Orthogonal Polynomials and Special Functions, SIAM Region Conference Series in Applied Mathematics, vol. 21. Philadelphia: SIAM.

    Book  Google Scholar 

  • Bombieri, E. 1967. On the local maximum property of the Koebe function. Inventiones Mathematicae 4: 26–67.

    Article  MathSciNet  MATH  Google Scholar 

  • Bombieri, E. 1967. Sulla seconda variazione della funzione di Koebe. Bollettino dell’Unione Matematica Italiana 22: 25–32.

    MathSciNet  MATH  Google Scholar 

  • Chihara, T. 1957. On co-recursive polynomials. Proceedings of the American Mathematical Society 8: 899–905.

    Article  MathSciNet  MATH  Google Scholar 

  • Greiner, R., and O. Roth. 2001. On support points of univalent functions and a disproof of a conjecture of Bombieri. Proceedings of the American Mathematical Society 129(12): 3657–3664.

    Article  MathSciNet  MATH  Google Scholar 

  • Ismail, M.E.H. 2005. Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics, vol. 98. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Killip, R., and B. Simon. 2003. Sum rules for Jacobi matrices and their applications to spectral theory. Annals of Mathematics 158: 253321.

    Article  MathSciNet  MATH  Google Scholar 

  • Leung, Y.J., and G. Schober. 1989. The simple-zero theorem of support points in \(\Sigma \). Proceedings of the American Mathematical Society 105(3): 603–608.

    MathSciNet  MATH  Google Scholar 

  • Peaslee, D.C., and W. Coppel. 1969. On an inequality of Bombieri. Journal of the Australian Mathematical Society 9(3–4): 399–404.

    Article  MathSciNet  MATH  Google Scholar 

  • Pommerenke, C. 1966. On the Loewner differential equation. The Michigan Mathematical Journal 13(4): 435–443.

    Article  MathSciNet  MATH  Google Scholar 

  • Pommerenke, C. 1975. Univalent Functions. Göttingen: Vandenhoeck and Ruprecht.

    MATH  Google Scholar 

  • Prokhorov, D., and O. Roth. 2004. On the local extremum property of the Koebe function. Mathematical Proceedings of the Cambridge Philosophical Society 136(2): 301–312.

    Article  MathSciNet  MATH  Google Scholar 

  • Prokhorov, D., and A. Vasil’ev. 2005. Optimal control in Bombieri’s and Tammi’s conjectures. Georgian Mathematical Journal 12(4): 743–761.

    MathSciNet  MATH  Google Scholar 

  • Schaeffer, A., and D. Spencer. 1950. Coefficient Regions for Schlicht Functions, American Mathematical Society Colloquium Publications, vol. 35, American Mathematical Society, New York City

  • Schaeffer, A., M. Schiffer, and D. Spencer. 1949. The coefficient regions of schlicht functions. Duke Mathematical Journal 16: 493–527.

    Article  MathSciNet  MATH  Google Scholar 

  • Slim, H. 1988. On co-recursive polynomials and their application to potential scattering. Journal of Mathematical Analysis and Applications 136: 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  • Stakgold, I., and M. Hoist. 2011. Green’s functions and boundary value problems, 3rd ed. New York: Wiley.

    Book  Google Scholar 

  • Szegő, G. 1975. Orthogonal polynomials, 4th edn., American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, Providence, RI

  • Van Assche, W. 1991. Orthogonal polynomials, associated polynomials and functions of the second kind. Journal of Computational and Applied Mathematics 37: 237–249.

    Article  MathSciNet  MATH  Google Scholar 

  • Van Assche, W. 1996. Compact Jacobi matrices: from Stieltjes to Krein and M(a, b). Annales de la Faculté des Sciences de Toulouse: Mathmatiques S5: 195–215.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee for many valuable comments.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, 19716, Newark, DE, USA

    Yuk-J. Leung

Authors
  1. Yuk-J. Leung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuk-J. Leung.

Additional information

Dedicated to Professor David Minda on the occasion of his retirement.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Leung, YJ. On the Bombieri numbers for the class S. J Anal 24, 229–250 (2016). https://doi.org/10.1007/s41478-016-0017-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-016-0017-2

Keywords

Mathematics Subject Classification

Search

Navigation