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Let  $  \{ {a _ {n} } : {1 \leq  n < \infty } \} $
be non-negative numbers, not all zero. Then

$$ \tag{1 }
( \sum a _ {n} )  ^ {4}  <  \pi  ^ {2} \sum a _ {n}  ^ {2}  \sum
n  ^ {2} a _ {n}  ^ {2} .
$$

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====
<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>