Growth and distortion theorems for slice monogenic functions

Abstract

Abstract. The sharp growth and distortion theorems are established for slice monogenic extensions of univalent functions on the unit disc D ⊂ C in the setting of Clifford algebras, based on a new convex combination identity. The analogous results are also valid in the quaternionic setting for slice regular functions and we can even prove the Koebe type one-quarter theorem in this case. Our growth and distortion theorems for slice regular (slice monogenic) extensions to higher dimensions of univalent holomorphic functions hold without extra geometric assumptions, in contrast to the setting of several complex variables in which the growth and distortion theorems fail in general and hold only for some subclasses with the starlike or convex assumption.

Growth Theorems in Slice Analysis of Several Variables

“…Based on a new convex combination identity in [18], the sharp growth theorems for slice monogenic extensions of univalent functions on the unit disc D ⊂ C in the setting of Clifford algebras was established as follows:…”

“…Let f be a slice monogenic function on the unit ball B in the regular quadratic cone R (m+1) (in this paper, R (m+1) denotes R m+1 in [18]) of R m such that its restriction f I to B I is injective and such that f (B I ) ⊆ C I for some I ∈ S m . If f (0) = 0, f ′ (0) = 1, then…”

“…(See [18] for details.) From the classical geometric function theory in higher dimensions, it is known that the growth theorems fail for the full class of normalized univalent mappings [15].…”

Slice Starlike Functions Over Quaternions

“…Based on Theorem B, the growth and distortion theorems were formulated for normalized injective slice regular functions in the special class N (B) (see [15,Theorem 3.11]). See [41] for further extensions to normalized injective (on BI ) slice regular functions in V(B).…”

“…In this paper, we shall answer an open problem in [41] and show that the class V(B) is not the largest subclass of slice regular functions such that the corresponding growth, distortion, and covering theorems hold. Indeed, it is proven that the growth, distortion, and covering theorems are valid in a tighter form for S * by applying a new growth theorem for the Carathéodory class in the quaternionic setting.…”