Ky Fan Inequality - Statement of The Classical Version

Statement of The Classical Version

If xi with 0 ≤ xi ≤ ½ for i = 1, ..., n are real numbers, then

\frac{ \bigl(\prod_{i=1}^n x_i\bigr)^{1/n} } { \bigl(\prod_{i=1}^n (1-x_i)\bigr)^{1/n} } \le \frac{ \frac1n \sum_{i=1}^n x_i } { \frac1n \sum_{i=1}^n (1-x_i) }

with equality if and only if x1 = x2 = . . . = xn.

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