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Schwarz-Pick Type Estimates for Gradients of Pluriharmonic Mappings of the Unit Ball

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Abstract

The main purpose of this paper is to find generalized Schwarz-Pick type inequalities for pluriharmonic functions f of the unit ball \({\mathbb {B}}^n\). The estimates for the gradient of |f| and f are given as follows:

$$\begin{aligned} \big |\nabla |f|(z)\big |\le \frac{4\sqrt{n}}{\pi (1-|z|)}, \ \ \ z\in {\mathbb {B}}^n \end{aligned}$$

and

$$\begin{aligned} |\nabla f(z)|^2\le \frac{2n(1-|f(z)|^2)}{1-|z|}, \ \ \ z\in {\mathbb {B}}^n. \end{aligned}$$

Moreover, by additional assuming the quasiregularity of f, some additional estimates are given in this paper.

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Acknowledgements

The author would like to thank the anonymous referees for a number of corrections and helpful suggestions on improving this paper.

Funding

The research of the author was supported by NSFs of China (No. 11501220), NSFs of Fujian Province (No. 2016J01020), and the Promotion Program for Young and Middle-aged Teachers in Science and Technology Research of Huaqiao University (ZQN-PY402).

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, People’s Republic of China

    Jian-Feng Zhu

  2. Department of Mathematics, Shantou University, Shantou, 515063, Guangdong, People’s Republic of China

    Jian-Feng Zhu

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  1. Jian-Feng Zhu
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Correspondence to Jian-Feng Zhu.

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Zhu, JF. Schwarz-Pick Type Estimates for Gradients of Pluriharmonic Mappings of the Unit Ball. Results Math 74, 114 (2019). https://doi.org/10.1007/s00025-019-1037-4

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