Carleman's inequality

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

:$sum\_\{n=1\}^infty\; left(a\_1\; a\_2\; cdots\; a\_n\; ight)^\{1/n\}\; le\; e\; sum\_\{n=1\}^infty\; a\_n.$

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

:$sum\_\{n=1\}^infty\; left\; (frac\{a\_1+a\_2+cdots\; +a\_n\}\{n\}\; ight\; )^ple\; left\; (frac\{p\}\{p-1\}\; ight\; )^psum\_\{n=1\}^infty\; a\_n^p$

for the non-negative numbers $a\_1,\; a\_2,\; a\_3,\; dots$ and $p>1,$ replacing each $a\_n$ with $a\_n^\{1/p\},$ and letting $p\; o\; infty.$

Carleman's inequality was first published in 1923 in a paper by Carleman [T. Carleman, "Sur les fonctions quasi-analytiques", Conférences faites au cinquième congres des mathématiciens Scandinaves, Helsinki (1923), 181-196.] ; it is used in the proof of Carleman's condition for the determinancy of the problem of moments.

Notes

References

*cite book
last = Hardy
first = G. H.
coauthors = Littlewood. J.E.; Pólya, G.
title = Inequalities, 2nd ed
publisher = Cambridge University Press
date = 1952
pages =
isbn = 0521358809

*cite book
last = Rassias
first = Thermistocles M., editor
title = Survey on classical inequalities
publisher = Kluwer Academic
date = 2000
pages =
isbn = 079236483X

*cite book
last = Hörmander
first = Lars
title = The analysis of linear partial differential operators I: distribution theory and Fourier analysis, 2nd ed
publisher = Springer
date = 1990
pages =
isbn = 354052343X

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