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Lipschitz Property of Harmonic Mappings with Respect to Pseudo-Hyperbolic Metric

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Abstract

In this paper, we show that harmonic Bloch mappings are Lipschitz continuous with respect to the pseudo-hyperbolic metric. This result improves the corresponding result of [11, Theorem 1]. Furthermore, we prove the similar property for harmonic quasiregular Bloch-type mappings.

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References

  1. J. Anderson, J. Clunie, and C. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math., 270 (1974), 12–37.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Sh. Chen, G. Liu, and A. Rasila, Characterizations of Lipschitz-type, Bloch-type and Dirichlet-type spaces, Monatsh. Math., 191 (2020), 513–536.

    Article  MathSciNet  MATH  Google Scholar 

  4. Sh. Chen, S. Ponnusamy, and A. Rasila, On characterizations of Bloch-type, Hardy-type and Lipschitz-type spaces, Math. Z., 279 (2015), 163–183.

    Article  MathSciNet  MATH  Google Scholar 

  5. Sh. Chen, S. Ponnusamy, and A. Rasila, Lengths, areas and Lipschitz-type spaces of planar harmonic mappings, Nonlinear Anal., 115 (2015), 62–70.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. H. Deng, S. Ponnusamy, and J. Qiao, Extreme points and support points of families of harmonic Bloch mappings, Potential Anal., (2020), DOI: https://doi.org/10.1007/s11118-020-09871-3.

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

    Book  MATH  Google Scholar 

  10. I. Efraimids, J. Gaona, R. Hernández, and O. Venegas, On harmonic Bloch-type mappings, Complex Var. Elliptic Equ., 62 (2017), 1081–1092.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Ghatage, J. Yan, and D. Zheng, Composition operators with closed range on the Bloch space, Proc. Amer. Math. Soc., 129 (2000), 2039–2044.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Huang, S. Ponnusamy, and J. Qiao, Extreme points and support points of harmonic α-Bloch mappings, Rocky Mountain J. Math., 50 (2020), 1324–1354.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Liu and S. Ponnusamy, On harmonic v-Bloch and v-Bloch-type mappings, Results Math., 73 (2018), Paper No. 90, 21 pp.

  15. C. Pommerenke, On Bloch functions, J. London Math. Soc., 2 (1970), 689–695.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer (Berlin, 1992).

    Book  MATH  Google Scholar 

  17. C. Xiong, On the Lipschitz continuity of the dilatation of Bloch functions, Period. Math. Hungar., 47 (2003), 233–238.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

We would like to thank the anonymous referee for helpful comments that had a significant impact on this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, China

    J. Huang & J.-F. Zhu

  2. Mathematics with Computer Science Program, Guangdong Technion, 241 Daxue Road, Jinping District, Shantou, Guangdong, 515063, People’s Republic of China

    J. Huang & A. Rasila

  3. Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 32000, Israel

    J. Huang & A. Rasila

Authors
  1. J. Huang
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  2. A. Rasila
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  3. J.-F. Zhu
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Corresponding author

Correspondence to A. Rasila.

Additional information

The research was supported by NNSF of China (No. 11501220, 11971182, 11971124), NSF of Fujian Province (No. 2021J01304, 2019J0101), and NSF of Guangdong Province (No. 2021A1515010326).

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Huang, J., Rasila, A. & Zhu, JF. Lipschitz Property of Harmonic Mappings with Respect to Pseudo-Hyperbolic Metric. Anal Math 48, 1069–1080 (2022). https://doi.org/10.1007/s10476-022-0132-z

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  • DOI: https://doi.org/10.1007/s10476-022-0132-z

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