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Difference between revisions of "Carleman inequality"

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The inequality
 
The inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204101.png" /></td> </tr></table>
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$$\sum_{n=1}^\infty(a_1\ldots a_n)^{1/n}<e\sum_{n=1}^\infty a_n$$
  
for arbitrary non-negative numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204102.png" />; discovered by T. Carleman [[#References|[1]]]. Here the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204103.png" /> can not be made smaller. The analogue of the Carleman inequality for integrals has the form:
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for arbitrary non-negative numbers $a_n\geq0$; discovered by T. Carleman [[#References|[1]]]. Here the constant $e$ can not be made smaller. The analogue of the Carleman inequality for integrals has the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204104.png" /></td> </tr></table>
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$$\int\limits_0^\infty\exp\left\lbrace\frac1x\int\limits_0^x\ln f(t)dt\right\rbrace dx<e\int\limits_0^\infty f(x)dx,\quad f(x)\geq0,$$
  
 
There are also other generalizations of the Carleman inequality, [[#References|[2]]].
 
There are also other generalizations of the Carleman inequality, [[#References|[2]]].
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====Comments====
 
====Comments====
The inequalities are strict, except for the trivial cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204105.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020410/c0204107.png" /> almost-everywhere.
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The inequalities are strict, except for the trivial cases $a_n=0$ for all $n$ and $f=0$ almost-everywhere.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. Bullen,  D.S. Mitrinović,  P.M. Vasić,  "Means and their inequalities" , Reidel  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. Bullen,  D.S. Mitrinović,  P.M. Vasić,  "Means and their inequalities" , Reidel  (1987)</TD></TR></table>

Revision as of 19:17, 31 July 2014

The inequality

$$\sum_{n=1}^\infty(a_1\ldots a_n)^{1/n}<e\sum_{n=1}^\infty a_n$$

for arbitrary non-negative numbers $a_n\geq0$; discovered by T. Carleman [1]. Here the constant $e$ can not be made smaller. The analogue of the Carleman inequality for integrals has the form:

$$\int\limits_0^\infty\exp\left\lbrace\frac1x\int\limits_0^x\ln f(t)dt\right\rbrace dx<e\int\limits_0^\infty f(x)dx,\quad f(x)\geq0,$$

There are also other generalizations of the Carleman inequality, [2].

References

[1] T. Carleman, , Wissenschaft. Vorträge 5. Kongress Skandinavischen Mathematiker , Helsinki (1923) pp. 181–196
[2] G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)


Comments

The inequalities are strict, except for the trivial cases $a_n=0$ for all $n$ and $f=0$ almost-everywhere.

References

[a1] P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1987)
How to Cite This Entry:
Carleman inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_inequality&oldid=32618
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article