Sierpiński’s inequality
by John from John D. Cook on (#6HMNT)
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:
When n > 2 we no longer have equality. However, W. Sierpiski, perhaps best known for the Sierpiski's triangle, proved that an inequality holds for all n. Given
we have the inequality
[1] W. Sierpinski. Sur une inegalite pour la moyenne alrithmetique, geometrique, et harmonique. Warsch. Sitzunsuber, 2 (1909), pp. 354-357.
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