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# The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$

Published:2015-02-06
Printed: Jun 2015
• Xiaomin Tang,
Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, P.R. China
• Taishun Liu,
Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, P.R. China
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## Abstract

Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n: |z_1|^{p_1}+|z_2|^{p_2}+\cdots+|z_n|^{p_2}\lt 1\}$ be an egg domain in $\mathbb{C}^n$. In this paper, we first characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq 1, p_2\geq 1)$, and then establish a new type of the classical boundary Schwarz lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic self-mappings of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta}, 0, \dots, 0)'$ and $\theta\in \mathbb{R}$.
 Keywords: holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain
 MSC Classifications: 32H02 - Holomorphic mappings, (holomorphic) embeddings and related questions 30C80 - Maximum principle; Schwarz's lemma, Lindelof principle, analogues and generalizations; subordination 32A30 - Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) {For functions of several hypercomplex variables, see 30G35}

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