Skip to main content
SpringerLink
Log in

Properties of normal harmonic mappings

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings.

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

Fig. 1

Similar content being viewed by others

References

  1. Ahlfors, L.: Complex Analysis. MacGraw-Hill, New York (1953)

    MATH  Google Scholar 

  2. Arbeláez, H., Hernández, R., Sierra, W.: Normal harmonic mappings. Monatsh Math. 190, 425–439 (2019)

    Google Scholar 

  3. Aulaskari, R., Makhmutov, S., Rättyä, J.: Results on meromorphic \(\varphi \)-normal functions. Complex Var. Elliptic Equ. 54, 855–863 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Colonna, F.: The Bloch constant of bounded harmonic mappings. Indiana Univ. Math. J. 38, 829–840 (1989)

    MathSciNet  MATH  Google Scholar 

  5. Duren, P.: Harmonic Mappings in the Plane. Cambridge University Press, New York (2004)

    MATH  Google Scholar 

  6. Hille, E.: Analytic Function Theory, vol. II. AMS Chelsea Publishing, New York (1962)

    MATH  Google Scholar 

  7. Lappan, P.: Some sequential properties of normal and non-normal functions with applications to automorphic functions. Comment. Math. Univ. St. Paul. 12, 41–57 (1964)

    MathSciNet  MATH  Google Scholar 

  8. Lappan, P.: A criterion for a meromorphic function to be normal. Comment. Math. Helv. 49, 492–495 (1974)

    MathSciNet  MATH  Google Scholar 

  9. Lappan, P.: The spherical derivative and normal functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 3, 301–310 (1977)

    MathSciNet  MATH  Google Scholar 

  10. Lehto, O., Virtanen, K.I.: Boundary behaviour and normal meromorphic functions. Acta Math. 97, 47–65 (1957)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Lohwater, A.J., Pommerenke, Ch.: On normal meromorphic functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 550, 12 (1973)

  13. Noshiro, K.: Contributions to the theory of meromorphic functions in the unit circle. J. Fac. Svi. Hokkaido Univ. 7, 149–159 (1938)

    MATH  Google Scholar 

  14. Pommerenke, C.: On normal and automorphic functions. Michigan Math. J. 193–202, (1974)

  15. Pommerenke, Ch.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)

  16. Yamashita, S.: On normal meromorphic functions. Math. Z. 141, 139–145 (1975)

    MathSciNet  MATH  Google Scholar 

  17. Yosida, K.: On a class of meromorphic functions. Proc. Phys. Math. Soc. Jpn. Ser. 16(3), 227–235 (1934)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hebei University, Baoding, 071002, Hebei, People’s Republic of China

    Hua Deng & Jinjing Qiao

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    Saminathan Ponnusamy

Authors
  1. Hua Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saminathan Ponnusamy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinjing Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinjing Qiao.

Additional information

Communicated by Adrian Constantin.

Publisher's Note

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

The research of this paper is supported by NSF of Hebei Science Foundation (No. A2018201033).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, H., Ponnusamy, S. & Qiao, J. Properties of normal harmonic mappings. Monatsh Math 193, 605–621 (2020). https://doi.org/10.1007/s00605-020-01459-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-020-01459-8

Keywords

Mathematics Subject Classification

Search

Navigation