SpringerLink
Log in

On Harmonic \({\varvec{\nu }}\)-Bloch and \({\varvec{\nu }}\)-Bloch-Type Map**s

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

The aim of this paper is twofold. One of them is to introduce the class of harmonic \(\nu \)-Bloch-type map**s as a generalization of harmonic \(\nu \)-Bloch map**s and thereby we generalize some recent results of harmonic 1-Bloch-type map**s investigated recently by Efraimidis et al. (Complex Var Elliptic Equ 62:1081–1092, 2017). The other is to investigate some subordination principles for harmonic Bloch map**s and then establish Bohr’s theorem for these map**s and in a general setting, in some cases.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abu Muhanna, Y.: Bohr’s phenomenon in subordination and bounded harmonic classes. Complex Variables Theory Appl. 55(11), 1071–1078 (2010)

    Article  MathSciNet  Google Scholar 

  2. Ali, R.M., Abu Muhanna, Y., Ponnusamy, S.: On the Bohr inequality, In: Progress in Approximation Theory and Applicable Complex Analysis (Edited by N. K. Govil et al.), Springer Optimization and Its Applications 117, 269–300 (2016)

  3. Anderson, J.M., Clunie, J., Pommerenke, Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12–37 (1974)

    MathSciNet  MATH  Google Scholar 

  4. Bonk, M., Minda, D., Yanagihara, H.: Distortion theorems for Bloch functions. Pacific. J. Math. 179, 241–262 (1997)

    Article  MathSciNet  Google Scholar 

  5. Chen, H., Gauthier, P.M., Hengartner, W.: Bloch constants for planar harmonic map**s. Proc. Amer. Math. Soc. 128(11), 3231–3240 (2000)

    Article  MathSciNet  Google Scholar 

  6. Chen, Sh, Ponnusamy, S., Wang, X.: Landau’s theorem and Marden constant for harmonic \(\nu \)-Bloch map**s. Bull. Aust. Math. Soc. 84, 19–32 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Chen, Sh, Ponnusamy, S., Wang, X.: Landau–Bloch constants for functions in \(\alpha \)-Bloch spaces and Hardy spaces. Complex Anal. Oper. Theory 6(5), 1025–1036 (2012)

    Article  MathSciNet  Google Scholar 

  8. Clunie, J.G., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A. I(9), 3–25 (1984)

    MathSciNet  MATH  Google Scholar 

  9. Chuaqui, M., Duren, P., Osgood, B.: The Schwarzian derivative for harmonic map**s. J. Anal. Math. 91, 329–351 (2003)

    Article  MathSciNet  Google Scholar 

  10. Colonna, F.: The Bloch constant of bounded harmonic map**s. Indiana Univ. Math. J. 38, 829–840 (1989)

    Article  MathSciNet  Google Scholar 

  11. Duren, P.: Harmonic Map**s in the Plane. Cambridge University Press, New York (2004)

    Book  Google Scholar 

  12. Efraimidis, I., Gaona, J., Hernández, R., Venegas, O.: On harmonic Bloch-type map**s. Complex Var. Elliptic Equ. 62, 1081–1092 (2017)

    Article  MathSciNet  Google Scholar 

  13. Garnett, J.B.: Bounded Analytic Functions, 1st edn. Springer, New York (2007)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Kayumov, I.R., Ponnusamy, S.: Bohr’s inequalities for the analytic functions with lacunary series and harmonic functions. J. Math. Anal. and Appl., 15 pages; (2018) https://doi.org/10.1016/j.jmaa.2018.05.038

    Article  MathSciNet  Google Scholar 

  16. Kayumov, I.R., Ponnusamy, S., Shakirov, N.: Bohr radius for locally univalent harmonic map**s, Math. Nachr, 12 pages; (2018) https://doi.org/10.1002/mana.201700068

    Article  Google Scholar 

  17. Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one map**s. Bull. Amer. Math. Soc. 42, 689–692 (1936)

    Article  MathSciNet  Google Scholar 

  18. Liu, G., Ponnusamy, S.: Uniformly locally univalent harmonic map**s associated with the pre-Schwarzian norm. Indagationes Mathematicae 29(2), 752–778 (2018)

    Article  MathSciNet  Google Scholar 

  19. Pommerenke, Ch.: On Bloch functions. J. Lond. Math. Soc. 2(2), 689–695 (1970)

    Article  MathSciNet  Google Scholar 

  20. Ponnusamy, S., Rasila, A.: Planar harmonic and quasiregular map**s. Topics in Modern Function Theory: Chapter in CMFT, RMS-Lecture Notes Series No. 19, 267–333 (2013)

  21. Schaubroeck, L.E.: Subordination of planar harmonic functions. Complex Variables 41, 163–178 (2000)

    MathSciNet  MATH  Google Scholar 

  22. Yamashita, S.: Schlicht holomorphic functions and the Riccati differential equation. Math. Z. 157, 19–22 (1977)

    Article  MathSciNet  Google Scholar 

  23. Zhu, J.: Coefficients estimate for harmonic \(\nu \)-Bloch map**s and harmonic \(K\)-quasiconformal map**s. Bull. Malays. Math. Sci. Soc. 39, 349–358 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Statistics (Hunan Provincial Key Laboratory of Intelligent Information Processing and Application), Hengyang Normal University, Hengyang, 421002, Hunan, People’s Republic of China

    Gang Liu

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

    Saminthan Ponnusamy

Authors
  1. Gang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saminthan Ponnusamy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Saminthan Ponnusamy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, G., Ponnusamy, S. On Harmonic \({\varvec{\nu }}\)-Bloch and \({\varvec{\nu }}\)-Bloch-Type Map**s. Results Math 73, 90 (2018). https://doi.org/10.1007/s00025-018-0853-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-018-0853-2

Keywords

Mathematics Subject Classification

Search

Navigation