Linear Invariance, Order and Convex Maps in

We continue the study, begun by the first author, of linear-invariant families in . We obtain the unexpected result that the Cayley transform (the analogue of the half-plane mapping in the complex plane) does not give the maximum or minimum distortion among all mappings of the unit ball of n≥ 2, onto convex domains. In addition a result analogous to that of Pommerenke that a linear invariant family has order 1 (the smallest possible order) if and only if it consists of convex mappings of the disk does not hold for the ball in n≥ 2. Finally, we extend these ideas to the polydisk in . The theory for this case bears some similarity to the theory for the ball but there are some striking differences as well. For example it is true that the family of convex holomorphic mappings of the polydisk in has minimum order (which is nin this case) but it is not true that for n>1 every linear-invariant family of minimum order consists of convex mappings.