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Carleman's inequality

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Theorem ([1], pp. 24) For positive real numbers $\{a_n\}_{n=1}^\infty$ , Carleman's inequality states that

$\displaystyle \sum_{n=1}^\infty \big(a_1 a_2 \cdots a_n\big)^{1/n} \le e \sum_{n=1}^\infty a_n.$

Although the constant

(the natural log base) is optimal, it is possible to refine Carleman's inequality by decreasing the weight coefficients on the right hand side [2].

References

1: L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
2: B.Q. Yuan, Refinements of Carleman's inequality, Journal of Inequalities in Pure and Applied Mathematics, Vol. 2, Issue 2, 2001, Article 21. online

"Carleman's inequality" is owned by Koro. [ owner history (1) ]

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Cross-references: right hand side, coefficients, weight, decreasing, natural log base, real numbers, positive
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This is version 2 of Carleman's inequality, born on 2003-06-26, modified 2003-12-20.
Object id is 4405, canonical name is CarlemansInequality.
Accessed 3402 times total.

Classification:

26D15 (Real functions :: Inequalities :: Inequalities for sums, series and integrals)

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