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A weighted version of Ozeki’s inequality. (English) Zbl 1031.26021

Authors’ abstract: “As an extension of Ozeki’s inequality [N. Ozeki, J. Coll. Arts Sci. Chiba Univ. 5(1968), No. 2, 199-203 (1969; MR 40#7408)] we give an inequality which estimates the difference \[ \sum^n_{k=1} p_k a^2_k \sum^n_{k=1} p_k b^2_k- \Biggl(\sum^n_{k=1} p_k a_k b_k\Biggr)^2 \] derived from the weighted Cauchy-Schwarz inequality for \(n\)-tuples \(a= (a_1,\dots, a_n)\), \(b= (b_1,\dots, b_n)\) and \(p= (p_1,\dots, p_n)\) be positive numbers under certain conditions. We discuss the upper bound of the difference not only in the general case but also in the special cases that \(a\) and \(b\) are monotonic in the opposite sense and in the same sense”.

MSC:

26D15 Inequalities for sums, series and integrals
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