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Let $\lbrace a_i \colon i=1\ldots n \rbrace$ be a set of $n$ integers greater than 1. Let $k$ be a positive integer. Is it true that:

$(\text{mean}(a_i))^k \leq \text{mean}(a_i^k)$


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You are basically asking whether $$\left(\frac{a_1+\dots+a_n}n\right)^k \le \frac{a_1^k+\dots+\dots a_n^k}n.$$

This is true if the function $f(x)=x^k$ is convex, it is the simplest form of Jensen's inequality.

The function $x^k$ is convex for $k\ge 1$, so the above is true for any positive integer. (And it is not important whether $a_i$'s are integers.)

Note that for $k=2$ you get $$\left(\frac{a_1+\dots+a_n}n\right)^2 \le \frac{a_1^2+\dots+\dots a_n^2}n$$ $$\frac{a_1+\dots+a_n}n \le \sqrt{\frac{a_1^2+\dots+\dots a_n^2}n}$$

This is the inequality between arithmetic and quadratic mean, see e.g. AoPS and wikipedia.

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