Elementary symmetric mean

The elementary symmetric mean is based on elementary symmetric polynomials. If $$x$$ is a tuple with $$x = (x_1,\dots,x_n)$$ and $$\alpha_p(x)$$ is the corresponding elementary symmetric term of order $$p$$, then the respective mean is
 * $$M_p(x) = \sqrt[p]{\frac{\alpha_p(x)}}$$.

Special cases

 * $$M_1(x)$$ is the arithmetic mean.
 * $$M_n(x)$$ is the geometric mean.