Carathéodory Theorems for Slice Regular Functions

In this paper a sharp quaternionic version of the Carathéodory theorem is established for slice regular functions with positive real part, which strengthes a weaken version recently established by 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.