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	'''Carleman's inequality''' is an [[inequality (mathematics)\|inequality]] in [[mathematics]], named after [[Torsten Carleman]], who proved it in 1923<ref>T. Carleman, ''Sur les fonctions quasi-analytiques'', Conférences faites au cinquième congres des mathématiciens Scandinaves, ~~Helsinki (1923), 181-196.</ref> and used it~~ to ~~prove the Denjoy–Carleman theorem on [[quasi-analytic]] classes.<ref>{{cite journal\|mr=2040885\|last1=Duncan\|first1=John\|last2=McGregor\|first2=Colin M.\|title=~~		26 yrs old Mechanical Executive Draftsperson Efrain Dorothy from Tisdale, loves to spend some time parachuting, como ganhar dinheiro na internet and urban exploration. Wants to travel and was stimulated after traveling to Guadalajara.<br><br>Also visit my blog post :: [http://comoficarrico.comoganhardinheiro101.com ganhando dinheiro na internet]
	Carleman's inequality\|journal=Amer. Math. Monthly \|volume=110\|year=2003\|issue=5\|pages= 424–431}}</ref><ref>{{cite journal\|mr=1820809\|last1=Pečarić\|first1=Josip\|last2=Stolarsky\|first2=Kenneth B.\|title=Carleman's inequality: history and new generalizations\|journal=Aequationes Math.\|volume=61\| year=2001\|issue=1–2\|pages=49–62}}</ref>

	~~==Statement==~~

	~~Let ''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, ... be a [[sequence]] of [[non-negative]] [[real number]]s, then~~

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

	The constant ''[[e (mathematical 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 some ~~element in the sequence is non-zero.~~

	~~==Integral version==~~

	~~Carleman's inequality has an integral version~~, ~~which states that~~

	~~:<math> \int_0^\infty \exp\left\{ \frac{1}{x} \int_0^x \ln f(t) dt \right\} dx \leq e \int_0^\infty f(x) dx </math>~~

	~~for any ''f'' ≥ 0~~.

	~~==Carleson's inequality==~~

	~~A generalisation, due~~ to [[Lennart Carleson]], states the following:<ref>{{cite journal\|first=L.\|last= Carleson\|title=A proof of an inequality of Carleman\|journal=Proc. Amer. Math. Soc.\|volume=5\|year=1954\|pages=932–933\|url=http://www.ams.org/journals/proc/1954-005-06/S0002-9939-1954-0065601-3/S0002-9939-1954-0065601-3.pdf}}</ref>

	~~for any convex function ''g'' with ''g''(0) = 0,~~ and ~~for any -1 < ''p'' < ∞,~~

	~~:<math> \int_0^\infty x^p e^{-g(x)/x} dx \leq e^{p+1} \int_0^\infty x^p e^{-g'(x)} dx. \,</math>~~

	~~Carleman's inequality follows from the case ''p'' = 0.~~

	~~==Proof==~~

	~~An elementary proof is sketched below. From the [[inequality of arithmetic and geometric means]] applied~~ to ~~the numbers <math>1\cdot a_1,2\cdot a_2,\dots,n \cdot a_n</math>~~

	~~:<math>\mathrm{MG}(a_1,\dots,a_n)=\mathrm{MG}(1a_1,2a_2,\dots,na_n)(n!)^{-1/n}\le \mathrm{MA}(1a_1,2a_2,\dots,na_n)(n!)^{-1/n}\, </math>~~

	~~where MG stands for geometric mean, and MA — for arithmetic mean~~. ~~The [[Stirling formula\|Stirling-type]] inequality <math>n!\ge \sqrt{2\pi n}\, n^n e^{-n}</math> applied to~~ <~~math~~>~~n+1~~<~~/math~~> ~~implies~~

	:~~<math>(n!)^{-1/n} \le \frac{e}{n+1}</math> for all <math>n\ge1.</math>~~

	~~Therefore~~

	:~~<math>MG(a_1,\dots,a_n) \le \frac{e}{n(n+1)}\, \sum_{1\le k \le n} k a_k \, ,</math>~~

	~~whence~~

	~~:<math>\sum_{n\ge1}MG(a_1,\dots,a_n) \le\, e\, \sum_{k\ge1} \bigg( \sum_{n\ge k} \frac{1}{n(n+1)}\bigg) \, k a_k =\, e\, \sum_{k\ge1}\, a_k \, ,</math>~~

	proving the inequality. Moreover, the inequality of arithmetic and geometric means of <math>n</math> non-negative numbers is known to be an equality if and only if all the numbers coincide, that is, in the present case, if and only if <math>a_k= C/k</math> for <math>k=1,\dots,n</math>. As a consequence, Carleman's inequality is never an equality for a convergent series, unless all <math>a_n</math> vanish, just because the [~~[harmonic series (mathematics)\|harmonic series]] is divergent.~~

	~~One can also prove Carleman's inequality by starting with [[Hardy's inequality]]~~

	:~~<math>\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</math>~~

	~~for the non-negative numbers ''a''<sub>1<~~/~~sub>,''a''<sub>2<~~/~~sub>,~~..~~. and ''p'' > 1, replacing each ''a''<sub>''n''</sub> with ''a''{{su\|b=n\|p=1/''p''}}, and letting ''p'' → ∞.~~

	~~==Notes==~~
	~~<references />~~

	~~==References==~~
	*{{cite book
	~~\| last = Hardy~~
	~~\| first = G. H.~~
	~~\| coauthors = Littlewood. J.E.; Pólya, G.~~
	~~\| title = Inequalities, 2nd ed~~
	~~\| publisher = Cambridge University Press~~
	~~\| year = 1952~~
	~~\| pages =~~
	~~\| isbn = 0-521-35880-9~~
	}}
	*{{cite book
	~~\| last = Rassias~~
	~~\| first = Thermistocles M., editor~~
	~~\| title = Survey on classical inequalities~~
	~~\| publisher = Kluwer Academic~~
	~~\| year = 2000~~
	~~\| pages =~~
	~~\| isbn = 0-7923-6483-X~~
	}}
	*{{cite book
	~~\| last = Hörmander~~
	~~\| first = Lars~~
	~~\| title = The analysis of linear partial differential operators I: distribution theory and Fourier analysis, 2nd ed~~
	~~\| publisher = Springer~~
	~~\| year = 1990~~
	~~\| pages =~~
	~~\| isbn = 3-540-52343-X~~
	}}

	~~==External links==~~
	* {{springer\|title=Carleman inequality\|id=p/c020410}}

	~~[[Category:Real analysis]]~~
	~~[[Category:Inequalities]~~]

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