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 (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.
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
- for all
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 a1/p
n, and letting p → ∞.
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