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A refinement of the Cauchy-Schwarz inequality. (English) Zbl 0763.26012

Summary: We prove: If \(x_ k\) and \(y_ k\) \((k=1,\dots,n)\) are real numbers satisfying \(0=x_ 0<x_ 1\leq x_ 2/2\leq\cdots\leq x_ n/n\) and \(0<y_ n\leq y_{n-1}\leq\cdots\leq y_ 1\), then \[ \left(\sum^ n_{k=1}x_ ky_ k\right)^ 2\leq\sum^ n_{k=1}y_ k\sum^ n_{k=1}\left(x^ 2_ k-{1 \over 4}x_ kx_{k-1}\right)y_ k \tag{*} \] with equality holding if and only if \(x_ k=kx_ 1\) \((k=1,\dots,n)\) and \(y_ 1=\cdots=y_ n\). Inequality \((*)\) is valid, in particular, if the sequence \((x_ k)\) is positive and convex.

MSC:

26D15 Inequalities for sums, series and integrals
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References:

[1] Bullen, P.S.; Mitrinović, D.S.; Vasić, P.M., Means and their inequalities, (1988), Reidel Dordrecht · Zbl 0687.26005
[2] McLaughlin, H.W., Inequalities complementary to the Cauchy-Schwarz inequality for finite sums of quaternions; refinements of the Cauchy-Schwarz inequality for finite sums of real numbers; inequalities concercing ratios and differences of generalized means of different order, University of maryland, technical note BN-454, (1966)
[3] Mitrinović, D.S., Analytic inequalities, (1970), Springer-Verlag New York · Zbl 0199.38101
[4] Ostrowski, A., ()
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