Extension operators and support points associated with g-Loewner chains on complex Banach spaces

Abstract

We prove that the Roper-Suffridge extension operator preserves g-parametric representation on general domains Bp1,p2={(z1,w)∈C×X:|z1|p1+‖w‖Xp2<1}, where X is a complex Banach space and p1,p2≥1. This result generalizes recent result from p1=2 to 1≤p1<∞. Next, we obtain a rigidity property of the Roper-Suffridge extension operator for the preservation of convexity. Finally, some sharp coefficient bounds for univalent mappings with g-parametric representation on the unit p-ball of Cn (1<p≤∞) are also obtained. As applications, we establish bounded support points with g-parametric representation on the unit p-ball. These results cover recent results related to bounded support points for families of biholomorphic mappings on the Euclidean unit ball for p=2, and on the unit polydisc for p=∞, respectively.