Distortion theorem for Bloch mappings on the unit ball . n

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f . H(. n) satisfying .f.0 = 1 and det f.(0) = . . (0, 1], where .f.0 = sup{(1 . |z|2)n+1/2n|det(f.(z))|1/n: z . . n.ub;. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f(0). When . = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.