Growth and distortion theorems for linearly invariant families on homogeneous unit balls in CnCn

Let BB be a homogeneous unit ball in X=CnX=Cn. In this paper, we obtain growth and distortion theorems for linearly invariant families FF of locally biholomorphic mappings on the unit ball BB with finite norm-order ‖ord‖e,1F. We use the Euclidean norm for the target space instead of the norm of XX, because we are able to obtain lower bounds in the two-point distortion theorems for linearly invariant families on any homogeneous unit ball in CnCn. We also obtain similar results for affine and linearly invariant families (A.L.I.F.s) of pluriharmonic mappings of the unit ball BB into CnCn. Again, in most of these results, we use the Euclidean norm for the target space, to obtain lower bounds in the two-point distortion theorems for A.L.I.F.s on BB. These results are generalizations to homogeneous unit balls of recent results due to Graham, Kohr and Pfaltzgraff, the authors of this paper, and Duren, Hamada and Kohr. In the last section, we consider two-point distortion theorems for L.I.F.s and A.L.I.F.s on the unit polydisc UnUn in CnCn.