Skip to main content
SpringerLink
Log in

Generalized support points of the set of univalent functions

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

The solutions to linear extremal problems, or the support points of the classS, have been extensively studied. Every support point is known to be a monotone slit mapping whose omitted arc is analytic. Our purpose here is to consider a more general class of continuous linear functionals onS and to investigate the geometric properties of their associated extremal functions. Under appropriate hypotheses we show that the generalized support points so produced are monotone slit mappings. We then give an example where the omitted set is not an analytic arc, thus proving the existence of an extreme point which is not a support point in the usual sense. Here we follow an idea of Hamilton [4], which he used for the same purpose, but our construction is simpler.

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. L. Brickman and D. Wilken,Support points of the set of univalent functions, Proc. Amer. Math. Soc.42 (1974), 523–528.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. L. Duren,Univalent Functions, Springer-Verlag, Heidelberg and New York, 1983.

    MATH  Google Scholar 

  3. J. Garnett,Analytic Capacity and Measure, Lecture Notes in Math. No. 297, Springer-Verlag, 1972.

  4. D. H. Hamilton,The extreme points of the class of schlicht functions, preprint, Univ. of Maryland, November 1983, 27 pp.; revision, January 1984, 30 pp.; revision under titleExtremal boundary problems for schlicht functions, April 1984, 71 pp.; revision, July 1984; partial revision under title,Extremal boundary problems for schlicht functions, I, February 1985, 36 pp.

  5. W. Hengartner and G. Schober,Some new properties of support points for compact families of univalent functions in the unit disk, Michigan Math. J.23 (1976), 207–216.

    Article  MATH  MathSciNet  Google Scholar 

  6. W. E. Kirwan and R. Pell,Extremal properties of a class of slit conformal mappings, Michigan Math. J.25 (1978), 223–232.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. I. Markushevich,Theory of Functions of a Complex Variable, Vol. III, Prentice-Hall, Englewood Cliffs, N.J., 1967.

    MATH  Google Scholar 

  8. Ch. Pommerenke,Univalent Functions, Vandenhoeck and Ruprecht, Göttingen, 1975.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI, USA

    Peter Duren & Y. J. Leung

  2. Department of Mathematics, University of Delaware, 19716, Newark, DE, USA

    Peter Duren & Y. J. Leung

Authors
  1. Peter Duren
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. J. Leung
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duren, P., Leung, Y.J. Generalized support points of the set of univalent functions. J. Anal. Math. 46, 94–108 (1986). https://doi.org/10.1007/BF02796576

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02796576

Keywords

Search

Navigation