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:

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

we have the inequality

## Related posts

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