Carathéodory Theorems for Slice Regular Functions

Abstract

In this paper a quaternionic sharp version of the Carathéodory theorem is established for slice regular functions with positive real part, which strengthes a weaken version recently established by D. Alpay et. al. using the Herglotz integral formula. Moreover, the restriction of positive real part can be relaxed so that the theorem becomes the quaternionic version of the Borel-Carathéodory theorem. It turns out that the two theorems are equivalent.2010 Mathematics Subject Classification. 30G35. 32A26.

Julia theory for slice regular functions

“…Here the last equality follows from equality (31). Now the proof of (i) is complete and it remains to prove (ii).…”

“…is an imaginary constant, say It 0 in virtue of the maximum principle for real part of regular functions (see Lemma 2 in [31]). A simple calculation shows that…”

Entire Slice Regular Functions

“…Function theory. The theory of slice regular functions was delevoped in the papers [47,104,106,107,109,110,112,113], in particular, the zeros were treated in [103,105,111] while further properties can be found in [33,44,85,92,93,102,140,141,150,151,152,154]. Slice monogenic functions with values in a Clifford algebra and their main properties were studied in [58,72,73,74,75,78,81,122,155].…”

Slice Starlike Functions Over Quaternions

“…From the assumption of f , we have As a direct consequence, we obtain all coefficient estimates for the Carathéodory class P applying the approach of finite average as in the case of one complex variable. Now let us recall this result in [39]. q n an be a slice regular function in P. Then…”