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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References:

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