Skip to main content
SpringerLink
Log in

Bergman space extremal problems for non-vanishing functions

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We consider the problem of minimising the Bergman Space A 2 norm of functions analytic and non-vanishing in the unit disc, which satisfy a finite number of constraints of the form l i (f) = c i , where each l i (f) is a finite linear combination of Taylor coefficients of f evaluated at certain points of the disc. We show that when the class of functions satisfying the constraints is nonempty, an extremal function exists and that every extremal function has rational outer part of a specific form.

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. Dov Aharonov, Catherine Bénéteau, Dmitry Khavinson, and Harold Shapiro, Extremal problems for nonvanishing functions in Bergman Spaces, Selected Topics in Complex Analysis, Birkhäuser, Basel, 2005, pp. 59–86.

    Google Scholar 

  2. D. Aharonov and H. S. Shapiro, A minimal area problem in conformal mapping, Part I, Preliminary Report, TRITA-MAT-1973-7, Royal Institute of Technology, Stockholm.

  3. D. Aharonov and H. S. Shapiro, A minimal area problem in conformal mapping, Part II, Preliminary Report, TRITA-MAT-1978-85, Royal Institute of Technology, Stockholm.

  4. T. Sheil-Small, Area extremal problems for non-vanishing functions, Proc. Amer. Math. Soc. 139 (2011), 3231–3245.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Sheil-Small, Complex Polynomials, Cambridge University Press, Cambridge, 2002.

    Book  MATH  Google Scholar 

  6. T. Sheil-Small, Solution to a Bergman space extremal problem for non-vanishing functions, Bull. London Math. Soc. 44 (2012), 763–774.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Ya. Khavinson, Foundations of the theory of extremal problems for bounded analytic functions with additional conditions, Two Papers on Extremal Problems in Complex Analysis, Amer. Math. Soc. Transl. Ser. 2, Vol. 129, 1986, pp. 63–114.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

    T. Sheil-Small

Authors
  1. T. Sheil-Small
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Sheil-Small.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sheil-Small, T. Bergman space extremal problems for non-vanishing functions. JAMA 118, 1–17 (2012). https://doi.org/10.1007/s11854-012-0027-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-012-0027-1

Keywords

Search

Navigation