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

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.