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.

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 (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.

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 ${\displaystyle 1\cdot a_{1},2\cdot a_{2},\dots ,n\cdot a_{n}}$

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