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

TL;DR

This paper investigates generalized power means related to Hardy's inequality, providing a comprehensive characterization of when these means satisfy Hardy-type inequalities and revealing their sensitive dependence on parameters.

Contribution

It offers a complete analysis of Hardy's inequalities for a class of generalized power means, highlighting their unique and delicate behavior.

Findings

Identifies parameter regions where Hardy inequalities hold for these means.

Shows that small parameter changes can invalidate Hardy inequalities for this family.

Reveals that this family behaves differently from other Hardy inequality-admitting means.

Abstract

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 (v_1,...,v_n) they take the q-th power means of all possible k-tuples (v_{i_1},...,v_{i_k}), and then calculate the s-th power mean of the resulting vector of length C_n^k. 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.