Skip to main content
SpringerLink
Log in

Distortion Theorems Associated with Schwarzian Derivative for Harmonic Mappings

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

Let f be a complex-valued harmonic mapping defined in the unit disc \({\mathbb {D}}\). The theorems of Chuaqui and Osgood (J Lond Math Soc 2:289–298, 1993), which assert that the bounds of the size of the hyperbolic norm of the Schwarzian derivative for an analytic function f imply certain bounds for distortion and growth of f, are extended to the harmonic case.

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.

Similar content being viewed by others

References

  1. Ahlfors, L.: Lectures on Quasiconformal Mappings. Van Nostrand, Princeton (1966)

    MATH  Google Scholar 

  2. Chuaqui, M., Osgood, B.: Sharp distortion theorems associated with the Schwarzian derivative. J. Lond. Math. Soc. 2, 289–298 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chuaqui, M., Hernández, R., Martín, M.J.: Affine and linear invariant families of harmonic mappings. Math. Ann. 367, 1099–1122 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Duren, P.L.: Harmonic Mapping in the Plane. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  5. Hernández, R., Martín, M.J.: Pre-Schwarzian and Schwarzian derivatives of harmonic mappings. J. Geom. Anal. 25, 64–91 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  7. Sheil-Small, T.: Constants for planar harmonic mappings. J. Lond. Math. Soc. 42, 237–248 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sobczak-Kneć, M., Starkov, V.V., Szynal, J.: Old and new order of linear invariant family of harmonic mappings and the bound for Jacobian. Ann. Univ. Mariae Curie-Sklodowska Sect. A 45, 191–202 (2011)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile

    Rodrigo Hernández

  2. Departamento de Ciencias Matemáticas y Físicas, Facultad de Ingeniería, Universidad Católica de Temuco, Av. Rudecindo Ortega 02950, Temuco, Chile

    Osvaldo Venegas

Authors
  1. Rodrigo Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Osvaldo Venegas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Osvaldo Venegas.

Additional information

Communicated by Daniel Aron Alpay.

The authors were partially supported by Fondecyt Grant # 1150284.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernández, R., Venegas, O. Distortion Theorems Associated with Schwarzian Derivative for Harmonic Mappings. Complex Anal. Oper. Theory 13, 1783–1793 (2019). https://doi.org/10.1007/s11785-018-0836-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-018-0836-z

Keywords

Mathematics Subject Classification

Search

Navigation