  We study relations between the Serre dimension defined as the growth of
entropy of the Serre functor and the global dimension of Bridgeland stability
conditions due to Ikeda-Qiu. A fundamental inequality between the Serre
dimension and the infimum of the global dimensions is proved. Moreover, we
characterize Gepner type stability conditions on fractional Calabi-Yau
categories via the Serre dimension, and classify triangulated categories of the
Serre dimension lower than one with a Gepner type stability condition.
