Do we know if $(\sum\limits_{i=1}^n a_i)^k \geq \sum\limits_{i=1}^n a_i^k$ for any $k\geq1$?
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If the $a_i$ are $\ge 0$, then the inequality is clear, since expanding $\left(\sum_1^n a_i\right)^k$ we get all the terms $a_i^k$, and also "cross-terms."
If some of the $a_i$ are allowed to be negative, let $k=n=2$. Let $a_1=1$ and $a_2=-1$. Then the left side is $0$ and the right side is $2$, so the inequality need not hold.
André Nicolas
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The case where $k$ is not an integer is not so straightforward. – Rodrigo Oct 28 '14 at 04:01
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1The Power Mean actually implies $LHS\le n^{k−1}RHS$ – Rodrigo Oct 28 '14 at 05:38
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1Does this theorem has a name? Is it a type of Jensen's Inequality? – luchonacho Mar 09 '17 at 11:45