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Harmonic mappings in Bergman spaces

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Abstract

In this paper, we investigate the properties of mappings in harmonic Bergman spaces. First, we discuss the coefficient estimate, the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit disk \(\mathbb D \) of \(\mathbb C \). Our results are generalizations of the corresponding ones in Chen et al. (Proc Am Math Soc 128:3231–3240, 2000), Chen et al. (J Math Anal Appl 373:102–110, 2011), Chen et al. (Ann Acad Sci Fenn Math 36:567–576, 2011). Then, we study the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit ball \(\mathbb B ^{n}\) of \(\mathbb C ^{n}\). The obtained results are generalizations of the corresponding ones in Chen and Gauthier (Proc Am Math Soc 139:583–595 2011). At last, we get a characterization for mappings in harmonic Bergman spaces on \(\mathbb B ^{n}\) in terms of their complex gradients.

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Acknowledgments

This research was partly supported by NSF of China (No. 11071063) and start project of Hengyang Normal University (No. 12334).

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Authors and Affiliations

  1. Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, 421008, Hunan, People’s Republic of China

    Sh. Chen

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600036, India

    S. Ponnusamy

  3. Department of Mathematics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Sh. Chen & X. Wang

Authors
  1. Sh. Chen
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  2. S. Ponnusamy
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  3. X. Wang
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Corresponding author

Correspondence to S. Ponnusamy.

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Communicated by A. Constantin.

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Chen, S., Ponnusamy, S. & Wang, X. Harmonic mappings in Bergman spaces. Monatsh Math 170, 325–342 (2013). https://doi.org/10.1007/s00605-012-0448-z

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