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On pluriharmonic \(\nu \)-Bloch-type mappings and hyperbolic-harmonic mappings

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Abstract

In this paper, we first establish a new version of Landau-type theorem of pluriharmonic mappings in the unit ball of \({\mathbb {R}}^{2n}\). Next we obtain a Bloch theorem of pluriharmonic \(\nu \)-Bloch-type mappings. Then, we provide a necessary condition for the hyperbolic-harmonic \(\nu \)-Bloch mappings in the unit ball of \({\mathbb {C}}^n\). Finally, we obtain a sufficient and necessary condition for the hyperbolic-harmonic \(\nu \)-Bloch mappings for the case of \(0<\nu \le 1\), which generalizes a result of Chen et al. (Math Model Anal 18:66–79, 2012).

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References

  1. 1.

    Aleman, A., Constantin, A.: Harmonic maps and ideal fluid flows. Arch. Ration. Mech. Anal. 204, 479–513 (2012)

  2. 2.

    Burgeth, B.: A Schwarz lemma for harmonic and hyperbolic-harmonic functions in higher dimensions. Manuscr. Math. 77(2–3), 283–291 (1992)

  3. 3.

    Chen, H.H., Gauthier, P.M., Hengartner, W.: Bloch constants for planar harmonic mappings. Proc. Am. Math. Soc. 128(11), 3231–3240 (2000)

  4. 4.

    Chen, H.H., Gauthier, P.M.: The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings. Proc. Am. Math. Soc. 139(2), 583–595 (2011)

  5. 5.

    Chen, S., Ponnusamy, S., Wang, X.: Properties of some classes of planar harmonic and planar biharmonic mappings. Complex Anal. Oper. Theory 5, 901–916 (2011)

  6. 6.

    Chen, S., Ponnusamy, S., Rasila, A.: Coefficient estimates, Landau’s theorem and Lipschitz-type spaces on planar harmonic mappings. J. Aust. Math. Soc. 96(2), 198–215 (2014)

  7. 7.

    Chen, S., Ponnusamy, S., Wang, X.: Coefficient estimates and Landau–Bloch’s theorem for planar harmonic mappings. Bull. Malays. Math. Sci. Soc. 34(2), 255–265 (2011)

  8. 8.

    Chen, S., Ponnusamy, S., Wang, X.: Landau’s theorem and Marden constant for harmonic \(\nu \)-Bloch mappings. Bull. Aust. Math. Soc. 84(1), 19–32 (2011)

  9. 9.

    Chen, S., Ponnusamy, S., Wang, X.: Weighted Lipschitz continuity, Schwarz–Pick’s lemma and Landau–Bloch’s theorem for hyperbolic-harmonic mappings in \({\mathbb{C}}^n\). Math. Model. Anal. 18(1), 66–79 (2012)

  10. 10.

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

  11. 11.

    Constantin, O., Martin, M.J.: A harmonic maps approach to fluid flows. Math. Ann. 369, 1–16 (2017)

  12. 12.

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

  13. 13.

    Dogan, O.F., Ersin, Üreyen A.: Weighted harmonic Bloch spaces on the ball. Complex Anal. Oper. Theory 12, 1143–1177 (2018)

  14. 14.

    Eriksson, S., Orelma, H.: A mean-value theorem for some eigenfunctions of the Laplace–Beltrami operator on the upper-half space. Ann. Acad. Sci. Fenn. Math. 36(1), 101–110 (2011)

  15. 15.

    Grigoryan, A.: Landau and Bloch theorems for harmonic mappings. Complex Var. Theory Appl. 51(1), 81–87 (2006)

  16. 16.

    Huang, X.Z.: Sharp estimate on univalent radius for planar harmonic mappings with bounded Fréchet derivative. Sci. Sin. Math. 44(6), 685–692 (2014). (in Chinese)

  17. 17.

    Liu, G., Ponnusamy, S.: On harmonic \(\nu \)-Bloch and \(\nu \)-Bloch-type mappings. Results Math. 73(3), 90 (2018)

  18. 18.

    Liu, M.S.: Estimates on Bloch constants for planar harmonic mappings. Sci. China Ser. A Math. 52(1), 87–93 (2009)

  19. 19.

    Liu, M.S.: Landau’s theorem for planar harmonic mappings. Computers Math. Appl. 57(7), 1142–1146 (2009)

  20. 20.

    Liu, M.S., Chen, H.H.: The Landau–Bloch type theorems for planar harmonic mappings with bounded dilation. J. Math. Anal. Appl. 468(2), 1066–1081 (2018)

  21. 21.

    Liu, M.S., Luo, L.F.: Landau-type theorems for certain bounded biharmonic mappings. Result Math. 74(4), 170 (2019)

  22. 22.

    Liu, M.S., Luo, L.F., Luo, X.: Landau–Bloch type theorems for strongly bounded harmonic mappings. Monatshefte für Mathematik (2019). https://doi.org/10.1007/s00605-019-01284-8

  23. 23.

    Liu, M.S., Yang, L.M.: Geometric properties and sections for certain subclasses of harmonic mappings. Monatsh. Math. 190(2), 353–387 (2019)

  24. 24.

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

  25. 25.

    Rudin, W.: Function Theory in the Unit Ball of \({\mathbb{C}}^n\). Springer, New York (1980)

  26. 26.

    Tam, L.F., Wan, T.Y.H.: On quasiconformal harmonic maps. Pac. J. Math. 53(2), 359–383 (1998)

  27. 27.

    Wood, J.C.: Lewys theorem fails in higher dimensions. Math. Scand. 69(2), 166–166 (1991)

  28. 28.

    Zhu, J.F.: Coefficients estimate for harmonic v-Bloch mappings and harmonic K-quasiconformal mappings. Bull. Malays. Math. Sci. Soc. 39(1), 349–358 (2016)

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Correspondence to Ming-Sheng Liu.

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This research is supported by Guangdong Natural Science Foundation (Grant No. 2018A030313508).

Communicated by Adrian Constantin.

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Xu, Z., Liu, M. On pluriharmonic \(\nu \)-Bloch-type mappings and hyperbolic-harmonic mappings. Monatsh Math (2020) doi:10.1007/s00605-019-01351-0

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Keywords

  • Harmonic mappings
  • Pluriharmonic mappings
  • \(\nu \)-Bloch-type mappings
  • Hyperbolic-harmonic \(\nu \)-Bloch mappings

Mathematics Subject Classification

  • Primary 31C10
  • Secondary 32A18
  • 31B05
  • 30C65