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

Lower and upper two-point distortion bounds for families F of biholomorphic mappings on the unit ball B of Cn 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.