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Sierpinski’s inequality. (English) Zbl 0693.26006

A simple proof is offered for the inequality in the title: \[ n^{n- 2}x_ 1x_ 2...x_ n\leq (x_ 1+x_ 2+...+x_ n)^{n-1}(x_ 1^{- 1}+x_ 2^{-1}+...+x_ n^{-1})\quad (x_ j>0,\quad j=1,2,...,n) \] with equality for \(n=1,2\) and for \(n=3\) equality holding iff \(x_ 1=x_ 2=...=x_ n.\)
Two estimations from above of the difference of the arithmetic and harmonic means are derived.
Reviewer: J.Aczél

MSC:

26D15 Inequalities for sums, series and integrals
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