Alzer, Horst A new refinement of the arithmetic mean – geometric mean inequality. (English) Zbl 0897.26004 Rocky Mt. J. Math. 27, No. 3, 663-667 (1997). The following inequality is offered as main result. Let \(x_{j}\) be positive reals, \(p_{j}\) \((j=1,\dots,n)\) nonnegative reals with sum 1, \(A\) the arithmetic, \(G\) the geometric mean of the \(x_{j}\) with weight \(p_{j}\) and \(M\) the maximum of the \(x_{j}\) \((j=1,\dots,n)\). Then \[ \sum_{j=1}^{n} p_{j}(x_{j}-G)^{2}\leq 2M(A-G) \] with equality iff all \(x_j\) with positive coefficients are equal. Reviewer: J.Aczél (Waterloo/Ontario) Cited in 1 ReviewCited in 10 Documents MSC: 26D15 Inequalities for sums, series and integrals Keywords:inequalities; arithmetic mean; maximum; geometric mean; weight PDFBibTeX XMLCite \textit{H. Alzer}, Rocky Mt. J. Math. 27, No. 3, 663--667 (1997; Zbl 0897.26004) Full Text: DOI Link References: [1] E.F. Beckenbach and R. Bellman, Inequalities , Springer Verlag, Berlin, 1983. [2] P.S. Bullen, D.S. Mitrinović and P.M. Vasić, Means and their inequalities , Reidel, Dordrecht, 1988. [3] D.I. Cartwright and M.J. Field, A refinement of the arithmetic mean-geometric mean inequality , Proc. Amer. Math. Soc. 71 (1978), 36-38. JSTOR: · Zbl 0392.26010 · doi:10.2307/2042211 [4] L. Grafakos, An elementary proof of the square summability of the discrete Hilbert transform , Amer. Math. Monthly 101 (1994), 456-458. JSTOR: · Zbl 0866.26014 · doi:10.2307/2974910 [5] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities , Cambridge University Press, Cambridge, 1954. [6] D.S. Mitrinović, Analytic inequalities , Springer Verlag, New York, 1970. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.