Roper-Suffridge extension operator and the lower bound for the distortion

Liczberski-Starkov gave a sharp lower bound for parallel toDphi(n)(f)(z)parallel to near the origin, where phi(n) is the Roper-Suffridge extension operator and f is a normalized convex mapping on the unit disk in C. They gave a conjecture that the sharp lower bound holds on the Euclidean unit ball B-n in C-n. In this paper, we will give a sharp lower bound on B-n for a more general extension operator and for normalized univalent mappings f or normalized convex mappings f. We will give a lower bound for mappings f in a linear invariant family. We will also give a similar sharp lower bound on bounded convex complete Reinhardt domains in C-n. (C) 2004 Elsevier Inc. All rights reserved.