Carath\'eodory theorems for Slice Regular Functions

In this paper a quaternionic sharp version of the Carath\'{e}odory 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-Carath\'{e}odory theorem. It turns out that the two theorems are equivalent.