Schwarzian derivatives and a linearly invariant family in ℂⁿ

We use Oda's definition of the Schwarzian derivative for locally univalent holomorphic maps F in several complex variables to define a Schwarzian derivative operator ΦF. We use the Bergman metric to define a norm ||φF|| for this operator, which in the ball is invariant under composition with automorphisms. We study the linearly invariant family estimating its order and norm order.