A Median Version of Hardy's Inequality

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 $f\in L^p(0,\mathrm{\infty })$, $p>1$, 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 $${\int }_0^{\mathrm{\infty }}\mathrm{\mid }M(t)-A(t){\mathrm{\mid }}^{p}dt\le 2^{1-p}{\left(\frac{p}{p-1}\right)}^p{\int }_0^{\mathrm{\infty }}f(t{)}^{p}dt,$$ 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.