Refinements of Ky Fan’s inequality

Abstract:

We prove the inequalities \[ A_n’ /G_n’ \leqslant (1 - G_n’ )/(1 - A_n’ ) \leqslant {A_n}/{G_n}\] and \[ A_n’ /G_n’ \leqslant (1 - {G_n})/(1 - {A_n}) \leqslant {A_n}/{G_n},\] where ${A_n}$ and ${G_n}$ (respectively, $A_n’$ and $G_n’$) denote the unweighted arithmetic and geometric means of ${x_1}, \ldots ,{x_n}$ (respectively, $1 - {x_1}, \ldots ,\;1 - {x_n}$) with ${x_i} \in (0,\tfrac {1} {2}](i = 1, \ldots ,n;n \geqslant 2$. Further we show that the ratios $(1 - G_n’ )/(1 - A_n’)$ and $(1 - {G_n})/(1 - {A_n})$ can be compared if and only if $n = 2$.