Homogeneous expansions of normalized biholomorphic convex mappings overB P

Abstract

The power series expansions of normalized biholomorphic convex mappings on the Reinhardt domain $$B^p = \left\{ z \right. \in \mathbb{C}^n :\left\| { z } \right\| _p = [\mathop \sum \limits_{j = 1} \left| {Z_j } \right| ^p ]^{1/p} 2 )$$ are studied. It is proved that the first (k+1) terms of the expansions of the jth componentf j of such a mapf depend only onz j , for 1 ⩽j⩽n, wherek is the natural number that satisfiesk < ρ ⩽k +I. Whenp→ ∞, this gives the result on the unit polydisc obtained by Suffridge in 1970.