Difference between revisions of "Carlson inequality"
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| + | $#C+1 = 7 : ~/encyclopedia/old_files/data/C020/C.0200460 Carlson inequality | ||
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| − | + | Let $ \{ {a _ {n} } : {1 \leq n < \infty } \} $ | |
| + | be non-negative numbers, not all zero. Then | ||
| − | + | $$ \tag{1 } | |
| + | ( \sum a _ {n} ) ^ {4} < \pi ^ {2} \sum a _ {n} ^ {2} \sum | ||
| + | n ^ {2} a _ {n} ^ {2} . | ||
| + | $$ | ||
| − | The | + | Proved by F. Carlson [[#References|[1]]]. The analogue of the Carlson inequality for integrals is: If $ f > 0 $, |
| + | $ f , x f \in L _ {2} ( 0 , \infty ) $, | ||
| + | then | ||
| + | |||
| + | $$ \tag{2 } | ||
| + | \left \{ \int\limits _ { 0 } ^ \infty | ||
| + | f (x) d x \right \} ^ {4} | ||
| + | \leq \pi ^ {2} \ | ||
| + | \left \{ \int\limits _ { 0 } ^ \infty | ||
| + | f ^ { 2 } (x) d x \right \} \ | ||
| + | \left \{ \int\limits _ { 0 } ^ \infty | ||
| + | x ^ {2} f ^ { 2 } (x) d x \right \} . | ||
| + | $$ | ||
| + | |||
| + | The constant $ \pi ^ {2} $ | ||
| + | is best possible in the sense that there exists a sequence $ \{ a _ {n} \} $ | ||
| + | such that right-hand side of (1) is arbitrarily close to the left-hand side, and there exists a function for which (2) holds with equality. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Carlson, "Une inegalité" ''Ark. Math. Astron. Fys.'' , '''25B''' : 1 (1934) pp. 1–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> F. Carlson, "Une inegalité" ''Ark. Math. Astron. Fys.'' , '''25B''' : 1 (1934) pp. 1–5</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)</TD></TR></table> | ||
Latest revision as of 21:32, 2 June 2020
Let $ \{ {a _ {n} } : {1 \leq n < \infty } \} $
be non-negative numbers, not all zero. Then
$$ \tag{1 } ( \sum a _ {n} ) ^ {4} < \pi ^ {2} \sum a _ {n} ^ {2} \sum n ^ {2} a _ {n} ^ {2} . $$
Proved by F. Carlson [1]. The analogue of the Carlson inequality for integrals is: If $ f > 0 $, $ f , x f \in L _ {2} ( 0 , \infty ) $, then
$$ \tag{2 } \left \{ \int\limits _ { 0 } ^ \infty f (x) d x \right \} ^ {4} \leq \pi ^ {2} \ \left \{ \int\limits _ { 0 } ^ \infty f ^ { 2 } (x) d x \right \} \ \left \{ \int\limits _ { 0 } ^ \infty x ^ {2} f ^ { 2 } (x) d x \right \} . $$
The constant $ \pi ^ {2} $ is best possible in the sense that there exists a sequence $ \{ a _ {n} \} $ such that right-hand side of (1) is arbitrarily close to the left-hand side, and there exists a function for which (2) holds with equality.
References
| [1] | F. Carlson, "Une inegalité" Ark. Math. Astron. Fys. , 25B : 1 (1934) pp. 1–5 |
| [2] | G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934) |
How to Cite This Entry:
Carlson inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carlson_inequality&oldid=17132
Carlson inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carlson_inequality&oldid=17132
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article