Minkowski's Inequality for Sums/Equality

From ProofWiki
Jump to navigation Jump to search

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


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


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

Retrieved from "https://proofwiki.org/w/index.php?title=Minkowski%27s_Inequality_for_Sums/Equality&oldid=744225"