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On the Cauchy-Schwarz inequality. (English) Zbl 0933.26010

The author proves that the inequality \[ \left( \sum _{k=1}^{n}x_{k}\cdot y_{k}\right)^{2} \leq\sum _{k=1}^{n} y_{k} \cdot \sum _{k=1} ^{n} \left( \alpha + \frac{\beta}{k} \right) \cdot x_{k}^2 \cdot y_k \] holds for all positive sequences \((y_{k})_{k=1}^{n}\) and all positive starshaped sequences \((x_{k})_{k=1}^{n}\) (as where named in reviewer’s paper [Math., Rev. Anal. Numér. Théor. Approximation, Anal. Numér. Théor. Approximation 12, 187-192 (1983; Zbl 0528.26005)] the sequences with the property \( 0 < x_{1} \leq x_{2}/2 \leq \cdots \leq x_{n} /n \)) if and only if \(\alpha \geq 3/4\) and \(\beta \geq 1- \alpha\). This result improves some inequalities given by Z. Liu [J. Math. Anal. Appl. 218, No. 1, 13-21 (1998; Zbl 0891.26011)]and the author [J. Math. Anal. Appl. 168, No. 2, 596-604 (1992; Zbl 0763.26012)].
Reviewer’s remark. Lemma 1, used in all these papers, was proven by the reviewer in the above mentioned paper.

MSC:

26D15 Inequalities for sums, series and integrals
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References:

[1] Alzer, H., A refinement of the Cauchy-Schwarz inequality, J. math. anal. appl., 168, 596-604, (1992) · Zbl 0763.26012
[2] Zheng, Liu, Remark on a refinement of the Cauchy-Schwarz inequality, J. math. anal. appl., 218, 13-21, (1998) · Zbl 0891.26011
[3] Mitrinović, D.S., Analytic inequalities, (1970), Springer-Verlag New York · Zbl 0199.38101
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