Ky Fan inequality

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In mathematics, the Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of numbers. If x_i, y_i for i = 1, ..., n are real numbers satisfying

$0 \le x_i \le\tfrac{1}{2}\,$ ,

$y_i=1-x_i\,$ ,

then

$\frac{ \left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}} } { \left(\prod_{i=1}^n y_i\right)^{\frac{1}{n}} } \leq \frac{ \frac{1}{n} \sum_{i=1}^n x_i } { \frac{1}{n} \sum_{i=1}^n y_i }.$

One method of proof uses the convexity of the functions involved on [0,1].

[edit] References

Neuman, Edward; Sándor, József (August 2005). "On the Ky Fan Inequality and Related Inequalities II". Bulletin of the Australian Mathematical Society 72 (1): 87–108. Australian Mathematical Publishing Assoc. Inc..