The order of a linearly invariant family in

We study the (trace) order of the linearly invariant family in the ball B n defined by ‖ S F ‖ ≤ α , where F : B n → C n is locally biholomorphic and S F is the Schwarzian operator. By adapting Pommerenke’s approach, we establish a characteristic equation for the extremal mapping that yields an upper bound for the order of the family in terms of α and the dimension n . Lower bounds for the order are established in similar terms by means of examples.