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 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 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

Direct 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.

By Hardy’s inequality

One can also prove Carleman's inequality by starting with Hardy's inequality
for the non-negative numbers,,… and, replacing each with, and letting.

Versions for specific sequences

Christian Axler and Mehdi Hassani investigated Carleman's inequality for the specific cases of where is the th prime number. They also investigated the case where. They found that if one can replace with in Carleman's inequality, but that if then remained the best possible constant.