Carleman's inequality explained

Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923^[1] and used it to prove the Denjoy - Carleman theorem on quasi-analytic classes.^{[2] [3]}

Statement

Let

a _{1,a} _{2,a} _{3,...} be a sequence of non-negative real numbers, then\sum ^{inf ty} _{n =1} \left(a _{1} a _{2} … a ^{1 /n} _{n\right)} \le e \sum ^{inf ty} _{n =1} a _{n.}

The constant

e (euler number) 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.

Integral version

Carleman's inequality has an integral version, which states that

\int ^{inf ty} _{0} \exp\left\{ (1)/(x) \int ^{x} _{0} ln f(t) d t \right\} d x \leq e \int ^{inf ty} _{0} f(x) d x

for any f ≥ 0.

Carleson's inequality

A generalisation, due to Lennart Carleson, states the following:^[4]

for any convex function g with g(0) = 0, and for any -1 < p < ∞,

\int ^{inf ty} _{0} x ^{p} e ^{-g(x)/x} d x \leq e ^{p+1} \int ^{inf ty} _{0} x ^{p} e ^{-g'(x)} d x.

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 ⋅ a _{1,2 ⋅} a _{2,...,n} ⋅ a _{n} MG (a _{1,...,a} _{n)=MG(1a} _{1,2a} _{2,...,na} ^{-1 /n} ^{n)(n!)} \le MA ( 1 a _{1,2a} _{2,...,na} ^{-1 /n} ^{n)(n!)}

where MG stands for geometric mean, and MA - for arithmetic mean. The Stirling-type inequality

n!\ge \sqrt{ 2 \pi n} n ^{n} e ^{-n} applied to n +1 implies(n!) ^{-1/n} \le (e)/(n +1) for all n\ge 1 .

Therefore,

MG(a _{1,...,a} _{n)} \le (e)/(n(n +1 )) \sum _{1\le} k a _{k} ,

whence

\sum _{n\ge1} MG(a _{1,...,a} _{n)} \le e \sum _{k\ge1} ( \sum _{n\ge} (1)/(n(n +1 )) ) k a _{k} = e \sum _{k\ge1} 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 , ... ,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 ^{inf ty} _{n =1} \left ( (a _{1+a} _{2+ …} + a _{n})/(n) \right ) ^{p\le} \left ( (p)/(p -1) \right ) ^{inf ty} _{n =1} a ^{p} _{n}

for the non-negative numbers a₁,a₂,... 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 = (1)/(p _{i})} .^[5] They found that if a _{i=p} _{i} one can replace e with (1)/(e) in Carleman's inequality, but that if a _{i = (1)/(p _{i})} then e remained the best possible constant.

Notes

1. T. Carleman, Sur les fonctions quasi-analytiques, Conférences faites au cinquième congres des mathématiciens Scandinaves, Helsinki (1923), 181-196.
2. 2040885. Duncan. John. McGregor. Colin M.. Carleman's inequality. Amer. Math. Monthly . 110. 2003. 5. 424 - 431. 10.2307/3647829.
3. 1820809. Pečarić. Josip. Stolarsky. Kenneth B.. Carleman's inequality: history and new generalizations. Aequationes Mathematicae. 61. 2001. 1 - 2. 49 - 62. 10.1007/s000100050160.
4. L.. Carleson. A proof of an inequality of Carleman. Proc. Amer. Math. Soc.. 5. 1954. 932 - 933. 10.1090/s0002-9939-1954-0065601-3. free.
5. Christian Axler, Medhi Hassani . Carleman's Inequality over prime numbers . Integers . 21, Article A53 . 13 November 2022.

References

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