  In this paper we investigate the inequality $S_q(\rho_{123})+S_q(\rho_2)\leq
S_q(\rho_{12})+S_q(\rho_{23}) \, (*)$ where $\rho_{123}$ is a state on a finite
dimensional Hilbert space $\mathcal{H}_1\otimes \mathcal{H}_2\otimes
\mathcal{H}_3,$ and $S_q$ is the Tsallis entropy. It is well-known that the
strong subadditivity of the von Neumnann entropy can be derived from the
monotonicity of the Umegaki relative entropy. Now, we present an equivalent
form of $(*)$, which is an inequality of relative quasi-entropies. We derive an
inequality of the form $S_q(\rho_{123})+S_q(\rho_2)\leq
S_q(\rho_{12})+S_q(\rho_{23})+f_q(\rho_{123})$, where $f_1(\rho_{123})=0$. Such
a result can be considered as a generalization of the strong subadditivity of
the von Neumnann entropy. One can see that $(*)$ does not hold in general (a
picturesque example is included in this paper), but we give a sufficient
condition for this inequality, as well.
