On some type of Hardy's inequality involving generalized power means

الأصل · EN

We discuss properties of certain generalization of Power Means proposed in 1971 by Carlson, Meany and Nelson. For any fixed parameter (k,s,q) and vector (v₁,...,vₙ) they take the q-th power means of all possible k-tuples (vᵢ₁,...,vᵢₖ), and then calculate the s-th power mean of the resulting vector of length Cₙᵏ. We work towards a complete answer to the question when such means satisfy inequalities resembling the classical Hardy inequality. We give a definitive answer in a large part of the parameter space. An unexpected corollary is that this family behaves much differently than most of other families admitting Hardy-type inequalities. Namely, arbitrarily small perturbations of parameters may lead to the breakdown of such inequalities.