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