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arXiv · 2605.25366

A Median Version of Hardy's Inequality

Gangsong Leng
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Abstract

Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average of $f$ over $(0,t)$, and let $M(t)$ be the lower median of $f$ over $(0,t)$. We show that \[ \int_0^\infty |M(t)-A(t)|^p\,dt \leq 2^{1-p}\left(\frac p{p-1}\right)^p \int_0^\infty f(t)^p\,dt, \] and that the constant is best possible. The proof is based on a pointwise rearrangement estimate coming from the half-measure property of the median, followed by the classical Hardy inequality. A discrete form and its sharpness are also included.

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Gangsong Leng. 2026-05-25. A Median Version of Hardy's Inequality. https://arxiv.org/abs/2605.25366

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