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The inequality of Ky Fan and related results

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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 numbersx i ε (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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  1. Morsbacher Str. 10, 51545, Waldbröl, Germany

    Horst Alzer

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  1. Horst Alzer
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Alzer, H. The inequality of Ky Fan and related results. Acta Appl Math 38, 305–354 (1995). https://doi.org/10.1007/BF00996150

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