On some properties of Tsallis hypoentropies and hypodivergences

Both the Kullback-Leibler and the Tsallis divergence have a strong limitation: if the value $0$ appears in probability distributions $\left( p_{1},\cdots ,p_{n}\right)$ and $\left( q_{1},\cdots ,q_{n}\right)$, it must appear in the same positions for the sake of significance. In order to avoid that limitation in the framework of Shannon statistics, Ferreri introduced in 1980 the hypoentropy: "such conditions rarely occur in practice". The aim of the present paper is to extend Ferreri's hypoentropy to the Tsallis statistics. We introduce the Tsallis hypoentropy and the Tsallis hypodivergence and describe their mathematical behavior. Fundamental properties like nonnegativity, monotonicity, the chain rule and subadditivity are established.