Szegő inequality

In functional analysis, a mathematical discipline, the Szegő inequality or Pólya–Szegő inequality, named after George Pólya and Gábor Szegő, states that if

$1\leq p<+\infty$

and

$u:\mathbb{R}^n\rightarrow \mathbb{R}^+\text{ in }W^{1,p}(\mathbb{R}^n),$

then

$\int_{\mathbb{R}^n} |\nabla u^*|^p \, d \mathcal{H}^n \leq \int_{\mathbb{R}^n} |\nabla u|^p \, d \mathcal{H}^n,$

where $u^*$ is the symmetric decreasing rearrangement of $u$.

See also

• Hölder's inequality