Partial subadditivity of entropies

In this paper a kind of partial subadditivity of the entropy is investigated in probability theory and in matrix theory. We show some new inequalities related to the subadditivity of the standard entropy and its one-parameter generalization in the discrete case. The matrix analogues of the inequalities are also studied. The standard entropy is related to the function -xlogx, but here some other functions are also used. A new version of the Tsallis entropy is included in the probabilistic case and a conjecture in the matrix case.