Let

with each component satisfying 0 < *x*_{i} ≤ 1/2. Define the complement *x*′ by taking the complement of each entry.

Let *G* and *A* represent the geometric and arithmetic mean respectively.

Then Ky Fan’s inequality says

Now let *H* be the harmonic mean. Since in general *H* ≤ *G* ≤ *A*, you might guess that Ky Fan’s inequality could be extended to

and indeed this is correct.

Source: Jósef Sándor. Theory and Means and Their Inequalities.