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

Liczberski–Starkov gave a sharp lower bound for DΦn(f )(z) near the origin, where Φn 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 Bn in Cn. In this paper, we will give a sharp lower bound on Bn 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 Cn.  2004 Elsevier Inc. All rights reserved.