Minkowski's Inequality for Sums/Corollary

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Corollary to Minkowski's Inequality for Sums

Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R$ be real numbers.

Let $p \in \R$ be a real number.


If $p > 1$, then:

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


Proof

\(\displaystyle \paren {\sum_{k \mathop = 1}^n \size {a_k + b_k}^p}^{1/p}\) \(\le\) \(\displaystyle \paren {\sum_{k \mathop = 1}^n \paren {\size {a_k} + \size {b_k} }^p}^{1/p}\) Triangle Inequality, $p > 0$
\(\displaystyle \) \(\le\) \(\displaystyle \paren {\sum_{k \mathop = 1}^n \size {a_k}^p}^{1/p} + \paren {\sum_{k \mathop = 1}^n \size {b_k}^p}^{1/p}\) Minkowski's Inequality for Sums

$\blacksquare$


Also known as

This result itself, like the main result of which it is referenced as a corollary, is sometimes called Minkowski's Inequality (for Sums).