Two-point distortion bounds for biholomorphic mappings of the ball in

Lower and upper two-point distortion bounds for families F of biholomorphic mappings on the unit ball B of C n are given in terms of the (trace) order of the linear-invariant family generated by F , bounds on ratios involving the derivative and Jacobian of the mappings in F , and the Carathéodory distance on B . (By two-point distortion bounds, we mean estimates on ‖ f ( b ) − f ( a ) ‖ for a , b ∈ B and f ∈ F .) This immediately results in growth bounds for such mappings. A contrast is drawn between these bounds and two-point distortion bounds in terms of the norm order of the generated linear-invariant family. As part of our work, we develop a lower distortion bound for automorphisms of B that may be of independent interest.