Carleman's inequality

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

${\displaystyle \sum _{n=1}^{\infty }\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}\leq e\sum _{n=1}^{\infty }a_{n}.}$

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 some element in the sequence is non-zero.

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

${\displaystyle \int _{0}^{\infty }\exp \left\{{\frac {1}{x}}\int _{0}^{x}\ln f(t)dt\right\}dx\leq e\int _{0}^{\infty }f(x)dx}$

for any f ≥ 0.

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

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

${\displaystyle \int _{0}^{\infty }x^{p}e^{-g(x)/x}dx\leq e^{p+1}\int _{0}^{\infty }x^{p}e^{-g'(x)}dx.}$

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

An elementary proof is sketched below. From the inequality of arithmetic and geometric means applied to the numbers ${\displaystyle 1\cdot a_{1},2\cdot a_{2},\dots ,n\cdot a_{n}}$

${\displaystyle \mathrm {MG} (a_{1},\dots ,a_{n})=\mathrm {MG} (1a_{1},2a_{2},\dots ,na_{n})(n!)^{-1/n}\leq \mathrm {MA} (1a_{1},2a_{2},\dots ,na_{n})(n!)^{-1/n}}$

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

${\displaystyle (n!)^{-1/n}\leq {\frac {e}{n+1}}}$ for all ${\displaystyle n\geq 1.}$

Therefore,

${\displaystyle MG(a_{1},\dots ,a_{n})\leq {\frac {e}{n(n+1)}}\,\sum _{1\leq k\leq n}ka_{k}\,,}$

whence

${\displaystyle \sum _{n\geq 1}MG(a_{1},\dots ,a_{n})\leq \,e\,\sum _{k\geq 1}{\bigg (}\sum _{n\geq k}{\frac {1}{n(n+1)}}{\bigg )}\,ka_{k}=\,e\,\sum _{k\geq 1}\,a_{k}\,,}$

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

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

${\displaystyle \sum _{n=1}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{p}\leq \left({\frac {p}{p-1}}\right)^{p}\sum _{n=1}^{\infty }a_{n}^{p}}$

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

• Hardy, G. H.; Littlewood J.E.; Pólya, G. (1952). Inequalities, 2nd ed. Cambridge University Press. ISBN 0-521-35880-9.
• Rassias, Thermistocles M., editor (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.

• Hazewinkel, Michiel, ed. (2001) [1994], "Carleman inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4