Carleman's inequality

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Carleman's inequality is an inequality in mathematics, named after Torsten Carleman. It states that if a_1, a_2, a_3, \dots is a sequence of non-negative real numbers, then

\sum_{n=1}^\infty \left(a_1 a_2 \cdots a_n\right)^{1/n} \le e \sum_{n=1}^\infty a_n.

The constant e in the inequality is optimal, that is, the inequality does not always hold if e is replaced by a smaller number. The inequality is strict (it holds with "<" instead of "≤") if all the elements in the sequence are positive.

One can prove Carleman's inequality by starting with Hardy's inequality

\sum_{n=1}^\infty \left (\frac{a_1+a_2+\cdots +a_n}{n}\right )^p\le \left (\frac{p}{p-1}\right )^p\sum_{n=1}^\infty a_n^p

for the non-negative numbers a_1, a_2, a_3, \dots and p > 1, replacing each an with a_n^{1/p}, and setting p\to \infty.

Carleman's inequality was first published in 1923 in a paper by Carleman[1]; it is used in the proof of Carleman's condition for the determinancy of the problem of moments.

[edit] Notes

  1. ^ T. Carleman, Sur les fonctions quasi-analytiques, Conférences faites au cinquième congres des mathématiciens Scandinaves, Helsinki (1923), 181-196.

[edit] References

  • Hardy, G. H.; Littlewood. J.E.; Pólya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0521358809. 
  • Rassias, Thermistocles M., editor (2000). Survey on classical inequalities. Kluwer Academic. ISBN 079236483X. 
  • Hörmander, Lars (1990). The analysis of linear partial differential operators I: distribution theory and Fourier analysis, 2nd ed. Springer. ISBN 354052343X. 

This article incorporates material from Carleman's inequality on PlanetMath, which is licensed under the GFDL.

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