Distortion Theorem for Bloch Mappings on the Unit Ball ■~n

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative tohyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem forBloch mappings f ∈ E(~n) satisfying ||f||= 1 and det f’(0) = α j (0,1], where ||f||= sup{(1 -|z|~2)~((n+1)/(2n))| det(f’(z))|: z ∈ ~n}. Here we establish the distortion theorem from a unified perspectiveand generalize some known results. This distortion theorem enables us to obtain a lower bound for theradius of the largest univalent ball in the image of f centered at f(0). When α = 1, the lower boundreduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides withthat of Bonk, Minda and Yanagihara.