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

Abstract

Liczberski–Starkov gave a sharp lower bound for ‖ D Φ n ( f ) ( z ) ‖ near the origin, where Φ 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 .