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The Schwarz Lemma at the Boundary of the Egg Domain Bp1,p2 in ℂn

Published online by Cambridge University Press:  20 November 2018

Xiaomin Tang  and
Taishun Liu
Xiaomin Tang
Affiliation:
Department of Mathematics, Huzhou University, Huzhou, Zhejiang hËh§§§, P.R. China. e-mail: [email protected] e-mail: [email protected]
Taishun Liu
Affiliation:
Department of Mathematics, Huzhou University, Huzhou, Zhejiang hËh§§§, P.R. China. e-mail: [email protected] e-mail: [email protected]
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Abstract

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Let ${{B}_{p1,p2}}\,=\,\left\{ z\,\in \,{{\mathbb{C}}^{n}}\,:\,{{\left| {{z}_{1}} \right|}^{{{p}_{1}}}}\,+\,{{\left| {{z}_{2}} \right|}^{{{p}_{2}}}}\,+\,\cdots \,+\,{{\left| {{z}_{n}} \right|}^{{{p}_{2}}}}\,<\,1 \right\}$ be an egg domain in ${{\mathbb{C}}^{n}}$. In this paper, we first characterize the Kobayashi metric on ${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge 1,\,{{p}_{2}}\,>\,1 \right)$ and then establish a new type of classical boundary Schwarz lemma at ${{z}_{0}}\in \partial {{B}_{{{p}_{1}},{{p}_{2}}}}$ for holomorphic self-mappings of ${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge \,1,\,{{p}_{2}}\,>\,1 \right)$), where ${{z}_{0}}={{\left( {{e}^{i\theta }},\,0,\ldots ,0 \right)}^{\prime }}$ and $\theta \,\in \,\mathbb{R}$.

Keywords

32H0230C8032A30holomorphic mappingSchwarz lemmaKobayashi metricegg domain
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 381 - 392
Copyright
Copyright © Canadian Mathematical Society 2015

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