Sierpiński's inequality for three classical means

Let A_n, G_n and H_n be the arithmetic mean, geometric mean, and harmonic mean of a set of n numbers.

When n = 2, the arithmetic mean times the harmonic mean is the geometric mean squared. The proof is simple:

$A_2(x, y) H_2(x, y) = \left(\frac{x + y}{2}\right)\left(\frac{2}{\frac{1}{x} + \frac{1}{y}} \right ) = xy = G_2(x,y)^2$

When n > 2 we no longer have equality. However, W. Sierpiński, perhaps best known for the Sierpiński’s triangle, proved that an inequality holds for all n. Given

$x = (x_1, x_2, \ldots, x_n)$

we have the inequality

$H_n(x)^{n-1}\, A_n(x) \leq G_n(x)^n \leq A_n(x)^{n-1}\, H_n(x)$

Newton’s inequality
Lehman’s inequality and circuits
The reversed Cauchy-Schwarz inequality

[1] W. Sierpinski. Sur une inégalité pour la moyenne alrithmétique, géometrique, et harmonique. Warsch. Sitzunsuber, 2 (1909), pp. 354–357.

Sierpiński’s inequality

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