On the integral approach to means and their Hardy property

Abstract

The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where $\mathcal{P}_p$ stands for the $p$-th power mean. One can ask about possible generalizations of this property to another families (with sharp constant depending on the mean). Adapting the notion of Riemann integral, for every weighted mean we define the lower and the upper integral mean. We prove that every symmetric, monotone, $\mathbb{R}$-weighted mean on $I$ which is continuous in its entries and weights has at most one continuous extension to the integral one. Moreover this extension preserves the Hardy constant. This result allows to extend the latter inequality to the family of homogeneous, concave deviation means.