Abstract
Letxi∈(0,1/2] (i=1,…,n) be real numbers. IfAnandGn(respectively,A′nandG′n) denote the weighted arithmetic and geometric means ofx1,…,xn(respectively, 1−x1,…,1−xn), then --Formula omitted -- with equality holding if and only ifx1=···=xn. The inequalities (*) provide refinements of Ky Fan's inequalityA′n/G′n≤An/Gnand its additive analogueA′n−G′n≤An−Gn.
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