Prescribing the Preschwarzian in several complex variables
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 $f\to f_α$ transform via the operator equation $[Df_α(z)]^{-1}D^2f_α(z)=α[Df(z)]^{-1}D^2f(z)$, and thereby, study properties of $f_α$. This is a natural generalization of the one variable operator $f_α(z)$ in \cite{DSS66} and the study of its univalence properties, e.g., the work of Royster \cite{Ro65} and many others. Möbius invariance and the multivariables Schwarzian derivative operator of T. Oda \cite{O} play a central role in this work.
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