Alzer, Horst Sierpinski’s inequality. (English) Zbl 0693.26006 Bull. Soc. Math. Belg., Sér. B 41, No. 2, 139-144 (1989). 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 Cited in 1 ReviewCited in 1 Document MSC: 26D15 Inequalities for sums, series and integrals Keywords:arithmetic mean; geometric mean; harmonic mean; Sierpinski’s inequality PDFBibTeX XMLCite \textit{H. Alzer}, Bull. Soc. Math. Belg., Sér. B 41, No. 2, 139--144 (1989; Zbl 0693.26006)