We solve the several complex variables preSchwarzian operator equation $[Df(z)]^{ -1}D^2f(z)=A(z)$, $z\in \C^n$, 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 $f\to f_\alpha$ transform via the operator equation $[Df_\alpha(z)]^{ -1}D^2f_\alpha(z)=\alpha[Df(z)]^{ -1}D^2f(z)$, and thereby, study properties of $f_\alpha$. This is a natural generalization of the one variable operator $f_\alpha(z)$ in \cite{ DSS66} and the study of its univalence properties, e.g., the work of Royster \cite{ Ro65} and many others. M\"{ o}bius invariance and the multivariables Schwarzian derivative operator of T. Oda \cite{ O} play a central role in this work.
Categories: math.CV