Prescribing the preSchwarzian in several complex variables

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 → f_α transform via the operator equation [Df_α(z)]^-1D²f_α(z) = a[Df(z)]^-1D²f(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.