Let An, Gn and Hn 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)

Related posts

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

The post Sierpiński’s inequality first appeared on John D. Cook.

Previous Post
Next Post