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 Denjoy–Carleman theorem on quasi-analytic classes.[2][3]

Contents

Statement

Let a1, a2, a3, ... be a sequence of non-negative real numbers, then

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

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 all the elements in the sequence are positive.

Integral version

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

\int_0^\infty \exp\left\{ \frac{1}{x} \int_0^x \ln f(t) dt \right\} dx \leq e \int_0^\infty f(x) dx

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_0^\infty x^p e^{-g(x)/x} dx \leq e^{p%2B1} \int_0^\infty x^p e^{-g'(x)} dx. \,

Carleman's inequality follows from the case p = 0.

Proof

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

\sum_{n=1}^\infty \left (\frac{a_1%2Ba_2%2B\cdots %2Ba_n}{n}\right )^p\le \left (\frac{p}{p-1}\right )^p\sum_{n=1}^\infty a_n^p

for the non-negative numbers a1,a2,... and p < 1, replacing each an with a1/p
n, and letting p → ∞.

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. MR2040885. 
  3. ^ Pečarić, Josip; Stolarsky, Kenneth B. (2001). "Carleman's inequality: history and new generalizations". Aequationes Math. 61 (1–2): 49–62. MR1820809. 
  4. ^ Carleson, L. (1954). "A proof of an inequality of Carleman". Proc. Amer. Math. Soc. 5: 932–933. 

References