We solve the several complex variables preSchwarzian operator equation [Df(z)]-1 D2f(z) = A(z), z € Cn, where A(z) is a bilinear operator and f is a Cn valued locally biholomorphic function on a domain in Cn. Then one can define a several variables f → fα transform via the operator equation [Dfα(z)]-1D2fα(z) = a[Df(z)]-1D2f(z), and thereby, study properties of fα. This is a natural generalization of the one variable operator fα(z) in  and the study of its univalence properties, e.g., the work of Royster  and many others. Möbius invariance and the multivariables Schwarzian derivative operator of Oda  play a central role in this work.
|Number of pages||10|
|Journal||Annales Academiae Scientiarum Fennicae Mathematica|
|State||Published - 2011|
- Holomorphic mapping
- PreSchwarzian derivative