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 \mathrm{e} (euler number) in the inequality is optimal, that is, the inequality does not always hold if \mathrm{e} is replaced by a smaller number. The inequality is strict (it holds with "<" instead of "≤") 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) = 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 n+1 implies Therefore, whence proving the inequality. Moreover, the inequality of arithmetic and geometric means of n 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 a_k= C/k for k=1,\dots,n. As a consequence, Carleman's inequality is never an equality for a convergent series, unless all a_n 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 a1/p n, and letting p → ∞.

Versions for specific sequences

Christian Axler and Mehdi Hassani investigated Carleman's inequality for the specific cases of a_i= p_i where p_i is the ith prime number. They also investigated the case where. They found that if a_i=p_i one can replace e with \frac{1}{e} in Carleman's inequality, but that if then e remained the best possible constant.