weighted power meanWeightedPowerMean2001-10-17 00:26:562005-01-30 15:49:36Definition\usepackage{amssymb}
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\usepackage{xypic}If $w_1,w_2,\ldots,w_n$ are positive real numbers such that $w_1+w_2+\cdots+w_n=1$, we define the \emph{$r$-th weighted power mean} of the $x_i$ as:
$$M_w^r(x_1,x_2,\ldots,x_n)=\left({w_1x_1^r+w_2x_2^r+\cdots+w_nx_n^r}\right)^{1/r}.$$
When all the $w_i=\frac{1}{n}$ we get the standard power mean.
The weighted power mean is a continuous function of $r$, and taking limit when $r\to0$ gives us
$$M_w^0=x_1^{w_1}x_2^{w_2}\cdots w_n^{w_n}.$$
We can weighted use power means to generalize the power means inequality:
If $w$ is a set of weights, and if $r<s$ then
$$M_w^r \leq M_w^s.$$