Abstract
In this survey paper, we present refinements, extensions, and variants of the inequality $$\mathop \Pi \limits_{i = 1}^n (x_i (1 - x_i ))^{1/n}< \sum\limits_{i = 1}^n {x_i } /\sum\limits_{i = 1}^n {(1 - x_i ),}$$ (*) valid for all real numbersxi e (0,1/2] (i=1,...,n) which are not all equal. Inequality (*) was published for the first time in 1961 and is due to Ky Fan.
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