Prescribing the Preschwarzian in several complex variables

Abstract

We solve the several complex variables preSchwarzian operator equation $[Df(z){]}^{-1}{D}^{2}f(z)=A(z)$, $z\in \C^n$, 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 $[D{f}_{\alpha }(z){]}^{-1}{D}^2{f}_{\alpha }(z)=\alpha [Df(z){]}^{-1}{D}^{2}f(z)$, 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. Möbius invariance and the multivariables Schwarzian derivative operator of T. Oda \cite{O} play a central role in this work.