Extension Operators that Preserve Geometric and Analytic Properties of Biholomorphic Mappings

Abstract

AbstractIn this survey we are concerned with certain extension operators which take a univalent function f on the unit disc U to a univalent mapping F from the Euclidean unit ball B n in \(\mathbb{C}^{n}\) into \(\mathbb{C}^{n}\), with the property that \(f(z_{1}) = F(z_{1},0)\). This subject began with the Roper–Suffridge extension operator, introduced in 1995, which has the property that if f is a convex function of U then F is a convex mapping of B n. We consider certain generalizations of the Roper–Suffridge extension operator. We show that these operators preserve the notion of g-Loewner chains, where \(g(\zeta ) = (1-\zeta )/(1 + (1 - 2\gamma )\zeta )\), | ζ | < 1 and γ ∈ (0, 1). As a consequence, the considered operators preserve certain geometric and analytic properties, such as g-parametric representation, starlikeness of order γ, spirallikeness of type δ and order γ, almost starlikeness of order δ and type γ.We use the method of Loewner chains to generate certain subclasses of normalized biholomorphic mappings on the Euclidean unit ball B n in \(\mathbb{C}^{n}\), which have interesting geometric characterizations. We obtain the characterization of g-starlike and g-spirallike mappings of type \(\alpha \in (-\pi /2,\pi /2)\), as well as of g-almost starlike mappings of order α ∈ [0, 1), by using g-Loewner chains. Also, we will show that, under certain assumptions, the mapping F(z) = P(z)z, z ∈ B n, has g-parametric representation on B n, where \(P: B^{n} \rightarrow \mathbb{C}\) is a holomorphic function such that P(0) = 1.KeywordsBiholomorphic mapping g-Loewner chain g-Parametric representation g-Starlike mappingLoewner chainParametric representationRoper–Suffridge extension operatorSpirallike mappingStarlike mappingSubordination2000 AMS Subject Classification:32H0230C45