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Minkowski Inequality

From AoPSWiki

The Minkowski Inequality states that if is a nonzero real number, then for any positive numbers , the following holds: \left(\sum_{j=1}^{m}\left(\sum_{i=1}^{n}a_{ij}^r\right)^{s/r}\right)^{1/s}\geq \left(\sum_{i=1}^{n}\left(\sum_{j=1}^{m}a_{ij}^s\right)^{r/s}\right)^{1/r}

Notice that if either or is zero, the inequality is equivalent to Holder's Inequality.

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