Skip to main content
Log in

On the Schwarz derivative, the Bloch space and the Dirichlet space

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

It is well known the connection between the growth of the Schwarzian with both the univalence [see Beardon and Gehring (Comment Math Helv 55: 50–64, 1980), Nehari (Bull Am Math Soc 55:545–551, 1949), Ovesea (Novi Sad J Math 26(1):69–76, 1996)] and the quasiconformal extension of the function [see Ahlfors and Weill (Proc Am Math Soc 13:975–978, 1962), Osgood (Old and new on the Schwarzian derivative, Quasiconformal mappings and analysis. Springer, New York, 1998)]. This work shows that previous relationships have geometrical interpretations when the Schwarzian is applied on the Bloch space and on the Dirichlet space. These interpretations are given in terms of a family of three-dimensional cones. Even more, these function spaces allow us to obtain Möbius invariant properties related to the norm induced by the Schwarzian among other consequences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

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. Ahlfors, L.: Complex Analysis. McGraw-Hill, New York (1979)

    MATH  Google Scholar 

  2. Ahlfors, L., Weill, G.: A uniqueness theorem for Beltramy equations. Proc. Am. Math. Soc. 13, 975–978 (1962)

    Article  Google Scholar 

  3. Beardon, A., Gehring, F.W.: Schwarzian derivatives, the poincaré metric and the kernel function. Comment. Math. Helv. 55, 50–64 (1980)

    Article  MathSciNet  Google Scholar 

  4. Danikas, N.: Some Banach spaces of analytic functions. Function Spaces and Complex Analysis, Univ. Joensuu Dept. Math. 2 University of Joensuu, Joensuu (1999)

  5. Kim, W.J.: The Schwarzian derivative and multivalence. Pac. J. Math. 31, 3 (1969)

    Article  MathSciNet  Google Scholar 

  6. Kummer, E.: Über die hypergeometrische Reihe. Crelle 15, 39–83 and 127–172 (1836)

  7. Nehari, Z.: The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc. 55, 545–551 (1949)

    Article  MathSciNet  Google Scholar 

  8. Osgood, B.: Old and new on the Schwarzian derivative, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995) Springer, New York, 275–276 (1998)

  9. Ovesea, H.: An univalence criterion and the Schwarzian derivative. Novi Sad J. Math. 26(1), 69–76 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Zhu, K.: Operator Theory in Function Spaces. 2nd ed, Mathematical Surveys and Monographs American Mathematical Society 138 (2007)

Download references

Acknowledgements

The author was partially supported by CONACYT.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to J. Oscar González Cervantes.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

González Cervantes, J.O. On the Schwarz derivative, the Bloch space and the Dirichlet space. Math Sci 14, 235–240 (2020). https://doi.org/10.1007/s40096-020-00334-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-020-00334-9

Keywords

Mathematics Subject Classification

Navigation