Carlson inequality

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)