Extremal Problems for Mappings with g-Parametric Representation on the Unit Polydisc in ℂ n

Abstract

In this paper we are concerned with extremal problems for mappings with g-parametric representation on the unit polydisc \(\mathbb {U}^2\) of \(\mathbb {C}^2\), where g is a univalent holomorphic function on the unit disc \(\mathbb {U}\) such that g(0) = 1, and which satisfies some natural conditions. In the first part of the paper, we obtain certain results related to extreme points and support points associated with the Caratheodory family \({\mathcal M}_g(\mathbb {U}^n)\) and the family \(S_g^*(\mathbb {U}^n)\) of g-starlike mappings on \(\mathbb {U}^n\). In particular, if g is a convex function on \(\mathbb {U}\), we use an analogue of the shearing process due to F. Bracci, to obtain sharp coefficient bounds for the family \({\mathcal M}_g(\mathbb {U}^2)\). In the last part of the paper, we are concerned with support points for the family \(S_g^0(\mathbb {U}^2)\) of mappings with g-parametric representation on \(\mathbb {U}^2\), where g is a convex function on \(\mathbb {U}\) with g(0) = 1, ℜg(ζ) > 0, \(\zeta \in \mathbb {U}\), and which satisfies certain natural conditions. Sharp coefficient bounds for the family \(S_g^0(\mathbb {U}^2)\), and various consequences and examples are obtained. Certain questions and conjectures are also formulated. This work complements recent work on extremal problems on the Euclidean unit ball \(\mathbb {B}^2\) in \(\mathbb {C}^2\).