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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 possiblegeneralizations of this property to another families (with sharp constantdepending on the mean).  Adapting the notion of Riemann integral, for every weighted mean we definethe lower and the upper integral mean. We prove that every symmetric, monotone,$\mathbb{R}$-weighted mean on $I$ which is continuous in its entries andweights has at most one continuous extension to the integral one. Moreover thisextension preserves the Hardy constant. This result allows to extend the latterinequality to the family of homogeneous, concave deviation means.

On the integral approach to means and their Hardy property