Schwarz lemma at the boundary on the classical domain of type ℐ𝒱

Let $R_{J V} (n)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="bold-script">ℛ</mi></mrow><mrow><mi mathvariant="bold-script">ℐ</mi><mi mathvariant="bold-script">V</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ be the classical domain of type $J V$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">ℐ</mi><mi mathvariant="bold-script">V</mi></math>$ in $C^{n}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math>$ with $n \geq 2$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi> <mo class="MathClass-rel">≥</mo> <mn>2</mn></math>$ . The purpose of this paper is twofold. The first is to investigate the boundary points of $R_{J V} (n)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="bold-script">ℛ</mi></mrow><mrow><mi mathvariant="bold-script">ℐ</mi><mi mathvariant="bold-script">V</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ . We give a sufficient and necessary condition such that the boundary points of $R_{J V} (n)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="bold-script">ℛ</mi></mrow><mrow><mi mathvariant="bold-script">ℐ</mi><mi mathvariant="bold-script">V</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ are smooth. The second is to establish the boundary Schwarz lemma on the classical domain of type $J V$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="bold-script">ℐ</mi><mi mathvariant="bold-script">V</mi></math>$ . we obtain the optimal estimates of the eigenvalues of the Fréchet derivative for holomorphic self-mappings at the smooth boundary point of $R_{J V} (n)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi mathvariant="bold-script">ℛ</mi></mrow><mrow><mi mathvariant="bold-script">ℐ</mi><mi mathvariant="bold-script">V</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>n</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ .

Keywords

holomorphic mapping, Schwarz lemma at the boundary, the classical domain of type

I V

$\mathcal{IV}$

Mathematical Subject Classification 2010

Primary: 32H02

Secondary: 32H99, 30C80

Milestones

Received: 2 May 2018

Revised: 13 August 2018

Accepted: 20 January 2019

Published: 5 November 2019

Authors

Jianfei Wang
School of Mathematical Sciences Huaqiao University Quanzhou, Fujian China

Taishun Liu
Department of Mathematics Huzhou Teachers College, Huzhou University Zhejiang China

Xiaomin Tang
Department of Mathematics Huzhou Teachers College, Huzhou University Zhejiang China

Vol. 302, No. 1, 2019

Jianfei Wang, Taishun Liu and Xiaomin Tang

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors