Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923[1] and used it to prove the Denjoy–Carleman theorem on quasi-analytic classes.
[2][3] Let
be a sequence of non-negative real numbers, then The constant
(euler number) in the inequality is optimal, that is, the inequality does not always hold if
is replaced by a smaller number.
The inequality is strict (it holds with "<" instead of "≤") if some element in the sequence is non-zero.
Carleman's inequality has an integral version, which states that for any f ≥ 0.
A generalisation, due to Lennart Carleson, states the following:[4] for any convex function g with g(0) = 0, and for any -1 < p < ∞, Carleman's inequality follows from the case p = 0.
An elementary proof is sketched below.
From the inequality of arithmetic and geometric means applied to the numbers
where MG stands for geometric mean, and MA — for arithmetic mean.
The Stirling-type inequality
applied to
implies Therefore, whence proving the inequality.
Moreover, the inequality of arithmetic and geometric means of
non-negative numbers is known to be an equality if and only if all the numbers coincide, that is, in the present case, if and only if
As a consequence, Carleman's inequality is never an equality for a convergent series, unless all
vanish, just because the harmonic series is divergent.
One can also prove Carleman's inequality by starting with Hardy's inequality[5]: §334 for the non-negative numbers
, replacing each
, and letting
Christian Axler and Mehdi Hassani investigated Carleman's inequality for the specific cases of
th prime number.
They also investigated the case where
[6] They found that if
one can replace
in Carleman's inequality, but that if
remained the best possible constant.