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.
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 âÂÂ¥ 0.
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.
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 , ,⦠and , replacing each with , and letting .
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.