Minkowski's Inequality

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Theorem

Lebesgue Spaces

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $f, g: X \to \R$ be $p$-integrable, that is, elements of Lebesgue $p$-space $\map {\mathcal L^p} \mu$.


Then their pointwise sum $f + g: X \to \R$ is also $p$-integrable, and:

$\norm {f + g}_p \le \norm f_p + \norm g_p$

where $\norm {\, \cdot \, }_p$ denotes the $p$-seminorm.


Theorem for Sums

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers.

Let $p \in \R$, $p \ne 0$ be a real number.

If $p < 0$, then we require that $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be strictly positive.


If $p > 1$, then:

$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \le \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$


If $p < 1$, $p \ne 0$, then:

$\displaystyle \left({\sum_{k \mathop = 1}^n \left({a_k + b_k}\right)^p}\right)^{1/p} \ge \left({\sum_{k \mathop = 1}^n a_k^p}\right)^{1/p} + \left({\sum_{k \mathop = 1}^n b_k^p}\right)^{1/p}$


Theorem for Integrals

Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$.



$(1):\quad$ Let $p > 1$. Then:
$\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \le \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$
$(2):\quad$ Let $p < 1, p \ne 0$. Then:
$\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \ge \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$


Source of Name

This entry was named for Hermann Minkowski.