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A self-improvement to the Cauchy-Schwarz inequality. (English) Zbl 1416.60040

Summary: We present a self improvement to the Cauchy-Schwarz inequality, which in the probability case yields \[ [E(X Y)]^2 \leq E(X^2) E(Y^2) - \left(| E(X) | \sqrt{\text{Var}(Y)} - | E(Y) | \sqrt{\text{Var}(X)}\right)^2 \text{.} \] It is to be noted that the additional term to the inequality only involves the marginal first two moments for \(X\) and \(Y\), and not any joint property. We also provide the discrete improvement to the inequality.

MSC:

60E15 Inequalities; stochastic orderings
26D15 Inequalities for sums, series and integrals
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