Cohen ring

In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean space Rⁿ then

$\underset{{\mathbb{ℝ}}^n}{\int }f(x)g(x)dx\le \underset{{\mathbb{ℝ}}^n}{\int }{f}^{*}(x)g^{*}(x)dx$

where f^* and g^* are the symmetric decreasing rearrangements of f(x) and g(x), respectively.

Proof

From layer cake representation we have:

$f(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\chi }_{f(x)>r}dr$

$g(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\chi }_{g(x)>s}ds$

where ${\chi }_{f(x)>r}$ denotes the indicator function of the subset E _f given by

$E_f=\{x\in X:f(x)>r\}$

Analogously, ${\chi }_{g(x)>s}$ denotes the indicator function of the subset E _g given by

$E_g=\{x\in X:g(x)>s\}$

$\begin{array}{rl}\underset{{\mathbb{ℝ}}^n}{\int }f(x)g(x)dx& =\underset{{\mathbb{ℝ}}^n}{\int }\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{\chi }_{f(x)>r}{\chi }_{g(x)>s}drdsdx\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{{\mathbb{ℝ}}^n}{\int }{\chi }_{f(x)>r\cap g(x)>s}dxdrds\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\mu (\{f(x)>r\}\cap \{g(x)>s\})drds\\ & \le \underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\min (\mu (f(x)>r);\mu (g(x)>s))drds\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\min (\mu (f^{*}(x)>r);\mu (g^{*}(x)>s))drds\\ & =\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\mu (\{f^{\ast }(x)>r\}\cap \{g^{\ast }(x)>s\})drds\\ & =\underset{{\mathbb{ℝ}}^n}{\int }{f}^{*}(x)g^{*}(x)dx\end{array}$

See also

• Rearrangement inequality
• Chebyshev's sum inequality