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.


26D15 Inequalities for sums, series and integrals
Full Text: DOI


[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., ()
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.