Schwarzian derivatives for pluriharmonic mappings

Iason Efraimidis, Álvaro Ferrada-Salas, Rodrigo Hernández, Rodrigo Vargas

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1 Scopus citations


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.

Original languageEnglish
Article number124716
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - 1 Mar 2021


  • Pluriharmonic mapping
  • Pre-Schwarzian derivative
  • Schwarzian derivative


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