Minkowski's Inequality for Sums/Equality

Theorem

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.

Then equality in Minkowski's Inequality for Sums, that is:

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

holds if and only if, for all $k \in \closedint 1 n$:

$\dfrac {a_k} {b_k} = c$

for some $c \in \R_{>0}$.

Proof

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Also known as

Minkowski's Inequality for Sums is also known just as Minkowski's Inequality.

However, this can cause it to be confused with Minkowski's Inequality for Integrals, so will not be used in this context on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Source of Name

This entry was named for Hermann Minkowski.

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.2$ Inequalities: Minkowski's Inequality for Sums: $3.2.12$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 36$: Inequalities: $36.12$: Minkowski's Inequality