  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.
