  In this paper, we initiate the study of relative operator entropies and
Tsallis relative operator entropies in the setting of JB-algebras. We show that
the operator inequalities on relative operator entropies and Tsallis relative
operator entropies still hold in our settings. In addition, we improve the
lower and upper bounds of the relative operator $(\alpha, \beta)$-entropy
established by Nikoufar which refined the bounds for the relative operator
entropy obtained earlier by Fujii and Kamei.
