Carleman's inequality

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

Statement

Let be a sequence of non-negative real numbers, then

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

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 < ,

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

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

for all

Therefore,

whence

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  .

Versions for specific sequences

Christian Axler and Mehdi Hassani investigated Carleman's inequality for the specific cases of where is the th prime number. They also investigated the case where .[5] They found that if one can replace with in Carleman's inequality, but that if then 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. Duncan, John; McGregor, Colin M. (2003). "Carleman's inequality". Amer. Math. Monthly. 110 (5): 424–431. doi:10.2307/3647829. MR 2040885.
  3. Pečarić, Josip; Stolarsky, Kenneth B. (2001). "Carleman's inequality: history and new generalizations". Aequationes Mathematicae. 61 (1–2): 49–62. doi:10.1007/s000100050160. MR 1820809.
  4. Carleson, L. (1954). "A proof of an inequality of Carleman" (PDF). Proc. Amer. Math. Soc. 5: 932–933. doi:10.1090/s0002-9939-1954-0065601-3.
  5. Christian Axler, Medhi Hassani. "Carleman's Inequality over prime numbers" (PDF). Integers. 21, Article A53. Retrieved 13 November 2022.

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.
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