 Hardy's inequality

Hardy's inequality bounds averages or weighted functions by their original size or derivatives. For $p>1$ and $f\ge 0$, its integral form is ${\int }_0^{\infty }(x^{-1}{\int }_0^{x}f(t)dt{)}^{p}dx\le (p/(p-1){)}^p{\int }_0^{\infty }f(x{)}^{p}dx$. Spatial weighted forms include the Hardy inequality in Euclidean space.