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On inequalities for bounded univalent functions

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Abstract

We provide inequalities for conformal maps ϕ of the unit disc \(\mathbb{D}\) and determine the functions for which the corresponding equality is satisfied.

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References

  1. D. Aharonov, On pairs of functions and related classes, Duke Math. J, 40 (1973), 669–676.

    Article  MathSciNet  MATH  Google Scholar 

  2. Ju. E. Alenicyn, On univalent functions without common values in multiply connected domains, Soviet Math. Dokl. 7 (1966), 305–307.

    Google Scholar 

  3. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, FL, 1995.

    MATH  Google Scholar 

  4. C. C. Cowen and Ch. Pommerenke, Inequalities for the angular derivative of an analytic function in the unit disk, J. London Math. Soc (2) 26 (1982), 271–289.

    Article  MathSciNet  MATH  Google Scholar 

  5. P. L. Duren, Univalent Functions, Springer, New York, 1983

    MATH  Google Scholar 

  6. J. A. Hummel, Inequalities of Grunsky type for Aharonov pairs, J. Analyse Math. 25 (1972), 217–257.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. A. Jenkins, On Bieberbach–Eilenberg functions III, Trans. Amer. Math. Soc. 119 (1965), 195–215.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Ku¨hnau, U¨ ber die schlichte konforme Abbildung auf nicht u¨ber-lappende Gebiete, Math. Nachr. 36 (1968), 61–71.

    Article  MathSciNet  Google Scholar 

  9. N. A. Lebedev, Applications of the area principle to problems on non-overlapping regions, Trudy Mat. Inst. Steklov 60 (1961), 211–231.

    MathSciNet  Google Scholar 

  10. Z. Nehari, Some inequalities in the theory of functions, Trans. Amer. Math. Soc. 75 (1953), 256–286.

    Article  MathSciNet  MATH  Google Scholar 

  11. Z. Nehari, Inequalities for the coefficients of univalent functions, Arch. Rational Mech. Anal. 34 (1969), 301–330.

    Article  MathSciNet  MATH  Google Scholar 

  12. Ch. Pommerenke, Univalent Functions, Vandenhoeck and Ruprecht, Go¨ttingen, 1975.

    Google Scholar 

  13. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin 1992.

    Book  MATH  Google Scholar 

  14. Ch. Pommerenke and A. Vasil’ev, Angular derivatives of bounded univalent functions and extremal partitions of the unit disk, Pacific J. Math. 206 (2002), 425–450.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Schiffer and O. Tammi, On the coefficient problem for bounded univalent functions, Trans. Amer. Math. Soc. 140 (1969), 461–474.

    Article  MathSciNet  MATH  Google Scholar 

  16. G. Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid e ICMAT, 28040, Madrid, Spain

    Eva A. Gallardo-Gutiérrez

  2. Institut für Mathematik, MA 8–2, Technische Universität, D-10623, Berlin, Germany

    Christian Pommerenke

Authors
  1. Eva A. Gallardo-Gutiérrez
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  2. Christian Pommerenke
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Corresponding author

Correspondence to Eva A. Gallardo-Gutiérrez.

Additional information

Authors are partially supported by Plan Nacional I+D grants MTM2013-42105-P and MTM2016-77710-P

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Gallardo-Gutiérrez, E.A., Pommerenke, C. On inequalities for bounded univalent functions. JAMA 138, 673–685 (2019). https://doi.org/10.1007/s11854-019-0041-7

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  • DOI: https://doi.org/10.1007/s11854-019-0041-7

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