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 [math]\displaystyle{ a_1,a_2,a_3,\dots }[/math] be a sequence of non-negative real numbers, then

[math]\displaystyle{ \sum_{n=1}^\infty \left(a_1 a_2 \cdots a_n\right)^{1/n} \le \mathrm{e} \sum_{n=1}^\infty a_n. }[/math]

The constant [math]\displaystyle{ \mathrm{e} }[/math] (euler number) in the inequality is optimal, that is, the inequality does not always hold if [math]\displaystyle{ \mathrm{e} }[/math] 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

[math]\displaystyle{ \int_0^\infty \exp\left\{ \frac{1}{x} \int_0^x \ln f(t) \,\mathrm{d}t \right\} \,\mathrm{d}x \leq \mathrm{e} \int_0^\infty f(x) \,\mathrm{d}x }[/math]

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 < ∞,

[math]\displaystyle{ \int_0^\infty x^p \mathrm{e}^{-g(x)/x} \,\mathrm{d}x \leq \mathrm{e}^{p+1} \int_0^\infty x^p \mathrm{e}^{-g'(x)} \,\mathrm{d}x. }[/math]

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 [math]\displaystyle{ 1\cdot a_1,2\cdot a_2,\dots,n \cdot a_n }[/math]

[math]\displaystyle{ \mathrm{MG}(a_1,\dots,a_n)=\mathrm{MG}(1a_1,2a_2,\dots,na_n)(n!)^{-1/n}\le \mathrm{MA}(1a_1,2a_2,\dots,na_n)(n!)^{-1/n} }[/math]

where MG stands for geometric mean, and MA — for arithmetic mean. The Stirling-type inequality [math]\displaystyle{ n!\ge \sqrt{2\pi n}\, n^n \mathrm{e}^{-n} }[/math] applied to [math]\displaystyle{ n+1 }[/math] implies

[math]\displaystyle{ (n!)^{-1/n} \le \frac{\mathrm{e}}{n+1} }[/math] for all [math]\displaystyle{ n\ge1. }[/math]

Therefore,

[math]\displaystyle{ MG(a_1,\dots,a_n) \le \frac{\mathrm{e}}{n(n+1)}\, \sum_{1\le k \le n} k a_k \, , }[/math]

whence

[math]\displaystyle{ \sum_{n\ge1}MG(a_1,\dots,a_n) \le\, \mathrm{e}\, \sum_{k\ge1} \bigg( \sum_{n\ge k} \frac{1}{n(n+1)}\bigg) \, k a_k =\, \mathrm{e}\, \sum_{k\ge1}\, a_k \, , }[/math]

proving the inequality. Moreover, the inequality of arithmetic and geometric means of [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ a_k= C/k }[/math] for [math]\displaystyle{ k=1,\dots,n }[/math]. As a consequence, Carleman's inequality is never an equality for a convergent series, unless all [math]\displaystyle{ a_n }[/math] vanish, just because the harmonic series is divergent.

One can also prove Carleman's inequality by starting with Hardy's inequality

[math]\displaystyle{ \sum_{n=1}^\infty \left (\frac{a_1+a_2+\cdots +a_n}{n}\right )^p\le \left (\frac{p}{p-1}\right )^p\sum_{n=1}^\infty a_n^p }[/math]

for the non-negative numbers a₁,a₂,... and p > 1, replacing each a_n with a1/pn, and letting p → ∞.

Versions for specific sequences

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

References

• Hardy, G. H.; Littlewood J.E.; Pólya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0-521-35880-9. 
• Rassias, Thermistocles M., ed (2000). Survey on classical inequalities. Kluwer Academic. ISBN 0-7923-6483-X. 
• Hörmander, Lars (1990). The analysis of linear partial differential operators I: distribution theory and Fourier analysis, 2nd ed. Springer. ISBN 3-540-52343-X. 

External links

• Hazewinkel, Michiel, ed. (2001), "Carleman inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/c020410