Carleman's inequality


Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923 and used it to prove the Denjoy-Carleman theorem on quasi-analytic classes.

Statement

Let a1, a2, a3,... be a sequence of non-negative real numbers, then
The constant e in the inequality is optimal, that is, the inequality does not always hold if e is replaced by a smaller number. The inequality is strict if some element in the sequence is non-zero.

Integral version

Carleman's inequality has an integral version, which states that
for any f ≥ 0.

Carleson's inequality

A generalisation, due to Lennart Carleson, states the following:
for any convex function g with g = 0, and for any -1 < p < ∞,
Carleman's inequality follows from the case p = 0.

Proof

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 for. 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
for the non-negative numbers a1,a2,... and p > 1, replacing each an with a, and letting p → ∞.