Pólya–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