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 $r$ 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 $$M_w^r \leq M_w^s.$$