  A chain rule and a subadditivity for the entropy of type $\beta$, which is
one of the nonadditive entropies, were derived by Z.Dar\'oczy. In this paper,
we study the further relations among Tsallis type entropies which are typical
nonadditive entropies. The chain rule is generalized by showing it for Tsallis
relative entropy and the nonadditive entropy. We show some inequalities related
to Tsallis entropies, especially the strong subadditivity for Tsallis type
entropies and the subadditivity for the nonadditive entropies. The
subadditivity and the strong subadditivity naturally lead to define Tsallis
mutual entropy and Tsallis conditional mutual entropy, respectively, and then
we show again chain rules for Tsallis mutual entropies. We give properties of
entropic distances in terms of Tsallis entropies. Finally we show
parametrically extended results based on information theory.
