with each component satisfying 0 < xi ≤ 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.