Schwarzian derivatives for pluriharmonic mappings

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:495

Article

Efraimidis, Iason¹  Ferrada-Salas, Alvaro²  Hernandez, Rodrigo³  Vargas, Rodrigo² 

[1] Texas Tech Univ, Dept Math & Stat, Box 41042, Lubbock, TX 79409 USA

[2] Pontificia Univ Catolica Chile, Fac Matemat, Santiago, Chile

[3] Univ Adolfo Ibanez, Fac Ingn & Ciencias, Av Padre Hurtado 750, Vina Del Mar, Chile

关键词: Pluriharmonic mapping;    Pre-Schwarzian derivative;    Schwarzian derivative;   

DOI  :  10.1016/j.jmaa.2020.124716

来源: Elsevier

【 摘 要 】

A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in Cn are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Mobius transformation. Some observations are made related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations, revealing differences between the theories in the plane and in higher dimensions. An example is given that rules out the possibility for a shear construction theorem to hold in Cn, for n >= 2. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free