Difference between revisions of "Carleman inequality"
From Encyclopedia of Mathematics
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The inequality | 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 | + | 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: |
| − | < | + | $$\int\limits_0^\infty\exp\left\lbrace\frac1x\int\limits_0^x\ln f(t)dt\right\rbrace dx<e\int\limits_0^\infty f(x)dx,\quad f(x)\geq0,$$ |
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 | + | 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=13636
Carleman inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_inequality&oldid=13636
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article