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A short proof of Ky Fan’s inequality. (English) Zbl 0759.26014

The famous Ky Fan inequality states that if \(A_ n\), \(A_ n'\) denote the arithmetic mean of \(x_ 1,\dots,x_ n\) and \(1-x_ 1,\dots,1-x_ n\), respectively, where \(x_ i\in(0,1/2]\) and \(G_ n\), \(G_ n'\) denote their geometric mean, then we have \(G_ n/G_ n'\leq A_ n/A_ n'\). The author, who obtained some remarkable relations connected to the Ky Fan inequality, in the present paper gives a new proof based on an identity of A. Dinghas [Math. Ann. 178, 315-334 (1968; Zbl 0162.078)].

MSC:

26D15 Inequalities for sums, series and integrals

Citations:

Zbl 0162.078
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