## Abstract

We solve the several complex variables preSchwarzian operator equation [Df(z)]^{-1} D^{2}f(z) = A(z), z € C^{n}, where A(z) is a bilinear operator and f is a C^{n} valued locally biholomorphic function on a domain in C^{n}. Then one can define a several variables f → f_{α} transform via the operator equation [Df_{α}(z)]^{-1}D^{2}f_{α}(z) = a[Df(z)]^{-1}D^{2}f(z), and thereby, study properties of f_{α}. This is a natural generalization of the one variable operator f_{α}(z) in [6] and the study of its univalence properties, e.g., the work of Royster [23] and many others. Möbius invariance and the multivariables Schwarzian derivative operator of Oda [17] play a central role in this work.

Original language | English |
---|---|

Pages (from-to) | 331-340 |

Number of pages | 10 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 36 |

Issue number | 1 |

DOIs | |

State | Published - 2011 |

## Keywords

- Holomorphic mapping
- PreSchwarzian derivative
- Univalence