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A new refinement of the arithmetic mean – geometric mean inequality. (English) Zbl 0897.26004

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.

MSC:

26D15 Inequalities for sums, series and integrals
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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.
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