Category:Minkowski's Inequality

This category contains pages concerning Minkowski's Inequality:

Lebesgue Spaces

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

Let $p \in \closedint 1 \infty$.

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

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:

$\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1/p} \le \paren {\sum_{k \mathop = 1}^n a_k^p}^{1/p} + \paren {\sum_{k \mathop = 1}^n b_k^p}^{1/p}$

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

$\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1/p} \ge \paren {\sum_{k \mathop = 1}^n a_k^p}^{1/p} + \paren {\sum_{k \mathop = 1}^n b_k^p}^{1/p}$

Theorem for Integrals

Let $f, g$ be (Darboux) integrable functions.

Let $p \in \R$ such that $p > 1$.

Then:

$\ds \paren {\int_a^b \size {\map f x + \map g x}^p \rd x}^{1/p} \le \paren {\int_a^b \size {\map f x}^p \rd x}^{1 / p} + \paren {\int_a^b \size {\map g x}^p \rd x}^{1 / p}$

Source of Name

This entry was named for Hermann Minkowski.