Mixed-mean inequalities for subsets

Abstract:

For $A\subset X=\{x_1,...,x_n ~|~x_i\geq 0, ~i=1,2,...,n\},$ let $a_A$ and $g_A$ denote the arithmetic mean and geometric mean of elements of $A$, respectively. It is proved that if $k$ is an integer in $(\frac {n}{2}, n]$, then \[ \Big (\prod _{|A|=k\atop A\subset X}a_{A}\Big )^{\frac {1}{C_n^k}} \geq \frac {1}{C_n^k}\Big (\sum _{|A|=k\atop A\subset X}g_{A}\Big ),\] with equality if and only if $x_1=...=x_n$. Furthermore, as a generalization of this inequality, a mixed power-mean inequality for subsets is established.