Skip to main content
SpringerLink
Log in

Generalized Grunsky coefficients and inequalities

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

The generalized Grunsky coefficients are defined in this paper for all locally univalent meromorphic functions in any domain in the complete complex plane. Various explicit formulas for these coefficients are established. Necessary conditions for univalence are obtained in arbitrary domains and in the unit disc in particular. The first one generalizes Grunsky inequalities and the second one is an extension of the Nehari-Schwarzian derivative condition.

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. D. Aharanov,The Schwarzian derivative and univalent functions, Doctoral Thesis, Technion, Israel Institute of Technology, 1967.

  2. D. Aharonov,A necessary and sufficient condition for univalence of a meromorphic function, Duke Math. J.36 (1969), 599–604.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Aharonov,Sequence of necessary conditions for univalence of a meromorphic function, Duke Math. J.38 (1971), 595–598.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Burbea,The Schwarzian derivative and the Poincaré metric, Pac. J. Math.85 (1979), 345–354.

    MATH  MathSciNet  Google Scholar 

  5. L. Comtet,Advanced Combinatorics, the Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, Boston, 1974.

    MATH  Google Scholar 

  6. H. W. Gould,Explicit formulas for Bernoulli numbers, Amer. Math. Monthly79 (1972), 44–51.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Harmelin,Aharonov invariants and univalent functions, Isr. J. Math.43 (1982), 244–254.

    MATH  MathSciNet  Google Scholar 

  8. R. Harmelin,Bergman kernel function and univalence criteria, J. Analyse Math.41 (1982), 249–258.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Hummel,The Grunsky coefficients of a Schlicht function, Proc. Amer. Math. Soc.15 (1964), 142–150.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Jabotinsky,Representation of functions by matrices, applications to Faber polynomials, Proc. Amer. Math. Soc.4 (1953), 546–553.

    Article  MATH  MathSciNet  Google Scholar 

  11. Ch. Pommerenke,Univalent Functions, Göttingen, 1975.

  12. P. G. Todorov,Explicit formulas for the coefficients of Faber polynomials with respect to univalent functions of the class Σ, Proc. Amer. Math. Soc.82 (1981), 431–438.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion — Israel Institute of Technology, Haifa, Israel

    Reuven Harmelin

Authors
  1. Reuven Harmelin
    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

Harmelin, R. Generalized Grunsky coefficients and inequalities. Israel J. Math. 57, 347–364 (1987). https://doi.org/10.1007/BF02766219

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02766219

Keywords

Search

Navigation