Minkowski's Inequality for Sums/Index Less than 1

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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 $p \in \R$ 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 $p < 1$, $p \ne 0$, then:

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



$q = \dfrac p {p - 1}$


$\dfrac 1 p + \dfrac 1 q = \dfrac 1 p + \dfrac {p - 1} p = 1$

In this case, $p$ and $q$ have opposite sign.

The proof then follows the same lines as the proof for $p > 1$, except that the Reverse Hölder's Inequality for Sums is applied instead.


Also known as

This result is also known as the Reverse Minkowski's Inequality (for Sums).

Source of Name

This entry was named for Hermann Minkowski.