Ky Fan’s inequality

Let

with each component satisfying 0 < x_i 1/2. Define the complement x' by taking the complement of each entry.

Let G andA 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: Josef Sandor. Theory and Means and Their Inequalities.

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