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Pinsker's and Fannes' type bounds on the Tsallis relative entropy arederived. The monotonicity property of the quantum $f$-divergence is used forits estimating from below. For order $\alpha\in(0,1)$, a family of lower boundsof Pinsker type is obtained. For $\alpha>1$ and the commutative case, uppercontinuity bounds on the relative entropy in terms of the minimal probabilityin its second argument are derived. Both the lower and upper bounds presentedare reformulated for the case of R\'{e}nyi's entropies. The Fano inequality isextended to Tsallis' entropies for all $\alpha>0$. The deduced bounds on theTsallis conditional entropy are used for obtaining inequalities of Fannes'type.

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy