On the Approximation of Univalent Functions by Subordinate Polynomials in the Unit Disk

  • Richard Greiner
  • Stephan Ruscheweyh
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


Let f be any conformal map in the unit disk D. We investigate the size of the largest number ρ = ρ(f, n) ∈ (0, 1], n ∈ ℕ such that there exists a univalent polynomial p n in D with f (0) = p n (0) and fD) ⊂ p n (D) ⊂ f(D). Clearly, these numbers are related to the quality of polynomial approximation to conformal maps. In this note we find a sharp uniform bound for ρ(f, n) if f is convex univalent, and discuss some related results and conjectures in the starlike and the general univalent case.


Unit Disk Univalent Function Maximal Range Extremal Function Univalent Polynomial 
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  1. [1]
    Andrievskij V., Ruscheweyh S. Maximal polynomial subordination to univalent functions in the unit disk. To appear in Constr. Approx.Google Scholar
  2. [2]
    Cordova A., Ruscheweyh S. On maximal ranges of polynomials spaces in the unit disk. Constr. Approx., 5: 309–327, 1989.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Cordova A., Ruscheweyh S. On maximal polynomial ranges in circular domains. Compl. Var., 10: 295–309, 1988.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Cordova A., Ruscheweyh S. On the maximal range problem for slit domains. In Computational Methods and Function Theory. Proceedings, Valparaiso, 1989. Lecture Notes in Mathematics. Springer-Verlag, 1990. S. Ruschweyh, E. B. Saff, L. C. Salinas, R. S. Varga.Google Scholar
  5. [5]
    Düren P. L. Univalent Functions. Springer-Verlag, Berlin, 1983.Google Scholar
  6. [6]
    Egervary E. Abbildungseigenschaften der arithmetischen Mittel der geometrischen Reihe. Math. Z., 42: 221–230, 1937.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Greiner R. Zur Güte der Approximation schlichter Abbildungen durch maximal subordinierende Polynomfolgen. Diplomarbeit, Würzburg, 1993.Google Scholar
  8. [8]
    Lewis J. Applications of a convolution theorem to Jacobi polynomials. SIAM Math. Anal., 10: 1110–1120, 1979.MATHCrossRefGoogle Scholar
  9. [9]
    Ruscheweyh, S. Geometric properties of Cesàro means. Results in Mathematics, 22: 739–748, 1992.MathSciNetMATHGoogle Scholar
  10. [10]
    Ruscheweyh S. Convolutions in Geometric Function Theory. Séminaire de Mathématiques supérieures, NATO Advanced Study Institute, Les Presses de l’Université de Montréal, Montréal, 1982.MATHGoogle Scholar
  11. [11]
    Ruscheweyh S., Salinas L. C. Subordination by Cesàro means. Compl. Var., 21: 279–285, 1993.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Richard Greiner
    • 1
  • Stephan Ruscheweyh
    • 1
  1. 1.Mathematisches InstitutUniversität WürzburgWürzburgGermany

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