关键词: distortion theorem, convex mapping, Roper-Suffridge operator

Abstract: Liczberski-Starkov first found a lower bound for ||D(f)|| near the origin, where f(z) = (F(z₁), F (z₁)z₂,… , F (z₁)z_n) is the Roper-Suffridge operator on the unit ball Bn in Cn and F is a normalized convex function on the unit disk. Later, Liczberski-Starkov and Hamada-Kohr proved the lower bound holds on the whole unit ball using a complex computation. Here we provide a rather short and easy proof for the lower bound. Similarly, when F is a normalized starlike function on the unit disk, a lower bound of ||D(f)|| is obtained again. 

Key words: distortion theorem, convex mapping, Roper-Suffridge operator