Ky Fan’s inequality

Let

$x = (x_1, x_2, x_3, \ldots, x_n)$

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

$x' = (1 - x_1, 1 - x_2, 1 - x_3, \ldots, 1 - x_n)$

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

Then Ky Fan’s inequality says

$\frac{G(x)}{G(x')} \leq \frac{A(x)}{A(x')}$

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

$\frac{H(x)}{H(x')} \leq \frac{G(x)}{G(x')} \leq \frac{A(x)}{A(x')}$

and indeed this is correct.

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