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

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

Mathematics Subject Classification

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