Estimating the Hardy constant of nonconcave Gini means

TL;DR

This paper investigates the Hardy constant for nonconcave Gini means, providing sharper upper bounds by constructing a majorizing mean for which the Hardy constant can be explicitly calculated.

Contribution

The paper introduces a method to estimate the Hardy constant for nonconcave Gini means by constructing a majorizing mean, advancing understanding of Hardy inequalities in this context.

Findings

Sharper upper bounds for Hardy constants of nonconcave Gini means

Construction of a majorizing mean with computable Hardy constant

Extension of Hardy inequality analysis to nonconcave Gini means

Abstract

The extension of the Hardy-Knopp-Carleman inequality to several classes of means was the subject of numerous papers. In the class of Gini means the Hardy property was characterized in 2015 by the second author. The precise value of the associated Hardy constant was only established for concave Gini means by the authors in 2016. The determination Hardy constant for nonconcave Gini means is still an open problem. The main goal of this paper is to establish sharper upper bounds for the Hardy constant in this case. The method is to construct a homogeneous and concave quasideviation mean which majorizes the nonconcave Gini mean and for which the Hardy constant can be computed.