Distortion theorem for Bloch mappings on the unit ball ℬ n

Abstract

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk \( \mathbb{D} \). This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H(ℬn) satisfying ‖f‖0 = 1 and det f′(0) = α ∈ (0, 1], where ‖f‖0 = sup{(1 − |z|2)n+1/2n|det(f′(z))|1/n: z ∈ ℬn⫂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 α = 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.