TY - JOUR

T1 - Schwarzian derivatives for pluriharmonic mappings

AU - Efraimidis, Iason

AU - Ferrada-Salas, Álvaro

AU - Hernández, Rodrigo

AU - Vargas, Rodrigo

N1 - Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - 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 Möbius 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.

AB - 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 Möbius 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.

KW - Pluriharmonic mapping

KW - Pre-Schwarzian derivative

KW - Schwarzian derivative

UR - http://www.scopus.com/inward/record.url?scp=85094649135&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2020.124716

DO - 10.1016/j.jmaa.2020.124716

M3 - Article

AN - SCOPUS:85094649135

SN - 0022-247X

VL - 495

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

M1 - 124716

ER -