More precisely, is the inequality $\Big(\displaystyle\sum_{n=1}^N a_n\Big)^p\leq\sum(a_n)^p$ true for $a_n\geq0$ for all $n\in\{1,\ldots,N\}$ and $p\in(0,\infty)$?
EDIT: And if so, will it also hold for an infinite sum under the same conditions?
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For $p>1$, see the answers to this question. – Lucian Oct 03 '14 at 00:06
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@Lucian: That question only addresses integers $p>1$. – Quang Hoang Oct 03 '14 at 01:23
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Neither direction on the inequality holds in general.
Consider $a_1 = a_2 = \ldots = a_n = 1$ and $p>1$, then
$$\left(\sum_{i=1}^n a_i \right)^p = n^p.$$
However,
$$\sum_{i=1}^n {a_i}^p = n < n^p$$
since $p>1$; thus,
$$\sum_{i=1}^n {a_i}^p < \left(\sum_{i=1}^n a_i\right)^p.$$
If instead $0<p<1$, then $n^p < n$, giving
$$\left(\sum_{i=1}^n a_i\right)^p < \sum_{i=1}^n {a_i}^p.$$
Cameron Williams
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Ok, so this doesn't hold in general if $p>1$, and you gave one example for when it does work when $0<p<1$ - does it always hold in this case, then? – PiTheMathemagician Oct 02 '14 at 23:57
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