Entropy inequalities for sums over several subsets and their applications to average entropy

For a set of random variables and a family of its subsets, Shearer´s inequality is well known as a generalization of the subadditivity of joint entropy. In this paper we generalize this inequality in case that each set of t random variables is contained in at least λ subsets. We also give a refinement of Han´s inequality on monotonicity of the average entropy.