Starlikeness for functions of a hypercomplex variable

On a Criterion of Local Invertibility and Conformality for Slice Regular Quaternionic Functions

“…We recall that in [5] the same Hermitian product of the Cullen and spherical derivatives of a slice regular function f appears in conditions which guarantee starlikeness for the function f .…”

Slice Starlike Functions Over Quaternions

“…In contrast to the wealthy results of geometric function theory for holomorphic starlike functions, there is essentially no result in the setting of quaternions except for [27]. One reason is that without the assumption preserving one slice, all tools such as the splitting lemma and the convex combination identity fail.…”

“…In fact, the algebraic condition in S * can also be described as a geometric restriction that the element in S * is a slice regular function for which its modulus is strictly increasing in the radial. Besides, we give some equivalent descriptions of the slice starlike function of order α (see Lemma 3.7) which allow us to present a new characterization of holomorphic starlike functions in one complex variable case (see Corollary 3.8) and find that the so-called slice starlike in [16,Definition 3.17] and algebraically starlike in [27,Definition 5.20] For the normalized and injective function f ∈ V(B), a quaternionic version of de Branges theorem was established and a natural question was raised if V(B) is the largest class of injective slice regular functions in which the Bieberbach conjecture holds [15].…”