The Roles Played by Order of Convexity or Starlikeness and the Bloch Condition in the Extension of Mappings from the Disk to the Ball

Abstract

Let \({G: \mathbb {C}^{n-1} \rightarrow \mathbb {C}}\) be holomorphic such that G(0) = 0 and DG(0) = 0. When f is a convex (resp. starlike) normalized (f(0) = 0, f′(0) = 1) univalent mapping of the unit disk \({\mathbb {D}}\) in \({\mathbb {C}}\) , then the extension of f to the Euclidean unit ball \({\mathbb {B}}\) in \({\mathbb {C}^n}\) given by \({\Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^{\prime}(z_1)} \, \hat{z}),\sqrt{f^{\prime}(z_1)}\, \hat{z})}\) , \({\hat{z}=(z_2,\dots,z_n) \in \mathbb {C}^{n-1}}\) , is known to be convex (resp. starlike) if G is a homogeneous polynomial of degree 2 with sufficiently small norm. Conversely, it is known that G cannot have terms of degree greater than 2 in its expansion about 0 in order for \({\Phi_G(f)}\) to be convex (resp. starlike), in general. We examine whether the restriction that f be either convex or starlike of a certain order \({\alpha \in (0,1]}\) allows, in general, for G to contain terms of degree greater than 2 and still have \({\Phi_G(f)}\) maintain the respective geometric property. Related extension results for convex and starlike Bloch mappings are also given.