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 $a_1,a_2,a_3,\dots$ be a sequence of non-negative real numbers, then

:$\sum_^\infty \left(a_1 a_2 \cdots a_n\right)^ \le \mathrm \sum_^\infty a_n.$

The constant $\mathrm$ (euler number) in the inequality is optimal, that is, the inequality does not always hold if $\mathrm$ 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

:$\int_0^\infty \exp\left\ \,\mathrmx \leq \mathrm \int_0^\infty f(x) \,\mathrmx$

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'' < ∞,

:$\int_0^\infty x^p \mathrm^ \,\mathrmx \leq \mathrm^ \int_0^\infty x^p \mathrm^ \,\mathrmx.$

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 $1\cdot a_1,2\cdot a_2,\dots,n \cdot a_n$

:$\mathrm(a_1,\dots,a_n)=\mathrm(1a_1,2a_2,\dots,na_n)(n!)^\le \mathrm(1a_1,2a_2,\dots,na_n)(n!)^$

where MG stands for geometric mean, and MA — for arithmetic mean. The Stirling-type inequality $n!\ge \sqrt\, n^n \mathrm^$ applied to $n+1$ implies

:$(n!)^ \le \frac$ for all $n\ge1.$

Therefore,

:$MG(a_1,\dots,a_n) \le \frac\, \sum_ k a_k \, ,$

whence

:$\sum_MG(a_1,\dots,a_n) \le\, \mathrm\, \sum_ \bigg( \sum_ \frac\bigg) \, k a_k =\, \mathrm\, \sum_\, a_k \, ,$

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

:$\sum_^\infty \left (\frac\right )^p\le \left (\frac\right )^p\sum_^\infty a_n^p$

for the non-negative numbers ''a''₁,''a''₂,... and ''p'' > 1, replacing each ''a''_''n'' with ''a'', 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 $i$th prime number, They also investigated the case where $a_i=\frac$. They found that if $a_i=p_i$ one can replace $e$ with $\frac$ in Carleman's inequality, but that if $a_i=\frac$ then $e$ remained the best possible constant.

Notes

References

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External links

* {{springer, title=Carleman inequality, id=p/c020410

Real analysis
Inequalities