#### Vol. 302, No. 1, 2019

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Schwarz lemma at the boundary on the classical domain of type $\mathcal{IV}$

### Jianfei Wang, Taishun Liu and Xiaomin Tang

Vol. 302 (2019), No. 1, 309–334
##### Abstract

Let ${\mathsc{ℛ}}_{\mathsc{ℐ}\mathsc{V}}\left(n\right)$ be the classical domain of type $\mathsc{ℐ}\mathsc{V}$ in ${ℂ}^{n}$ with $n\ge 2$. The purpose of this paper is twofold. The first is to investigate the boundary points of ${\mathsc{ℛ}}_{\mathsc{ℐ}\mathsc{V}}\left(n\right)$. We give a sufficient and necessary condition such that the boundary points of ${\mathsc{ℛ}}_{\mathsc{ℐ}\mathsc{V}}\left(n\right)$ are smooth. The second is to establish the boundary Schwarz lemma on the classical domain of type $\mathsc{ℐ}\mathsc{V}$. we obtain the optimal estimates of the eigenvalues of the Fréchet derivative for holomorphic self-mappings at the smooth boundary point of ${\mathsc{ℛ}}_{\mathsc{ℐ}\mathsc{V}}\left(n\right)$.

##### Keywords
holomorphic mapping, Schwarz lemma at the boundary, the classical domain of type $\mathcal{IV}$
##### Mathematical Subject Classification 2010
Primary: 32H02
Secondary: 32H99, 30C80