Carleman's inequality

Let $a\_1,a\_2,a\_3,\backslash dots$ be a sequence of non-negative real numbers, then

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash left(a\_1\; a\_2\; \backslash cdots\; a\_n\backslash right)^\{1/n\}\; \backslash le\; \backslash mathrm\{e\}\; \backslash sum\_\{n=1\}^\backslash infty\; a\_n.$

The constant $\backslash mathrm\{e\}$ (euler number) in the inequality is optimal, that is, the inequality does not always hold if $\backslash mathrm\{e\}$ 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

:$\backslash int\_0^\backslash infty\; \backslash exp\backslash left\backslash \{\; \backslash frac\{1\}\{x\}\; \backslash int\_0^x\; \backslash ln\; f(t)\; \backslash ,\backslash mathrm\{d\}t\; \backslash right\backslash \}\; \backslash ,\backslash mathrm\{d\}x\; \backslash leq\; \backslash mathrm\{e\}\; \backslash int\_0^\backslash infty\; f(x)\; \backslash ,\backslash mathrm\{d\}x$

for any f ≥ 0.

A generalisation, due to Lennart Carleson, states the following:{{cite journal|first=L.|last= Carleson|title=A proof of an inequality of Carleman|journal=Proc. Amer. Math. Soc.|volume=5|year=1954|pages=932–933|url=https://www.ams.org/journals/proc/1954-005-06/S0002-9939-1954-0065601-3/S0002-9939-1954-0065601-3.pdf|doi=10.1090/s0002-9939-1954-0065601-3|doi-access=free}}

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

:$\backslash int\_0^\backslash infty\; x^p\; \backslash mathrm\{e\}^\{-g(x)/x\}\; \backslash ,\backslash mathrm\{d\}x\; \backslash leq\; \backslash mathrm\{e\}^\{p+1\}\; \backslash int\_0^\backslash infty\; x^p\; \backslash mathrm\{e\}^\{-g\text{'}(x)\}\; \backslash ,\backslash mathrm\{d\}x.$

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 $1\backslash cdot\; a\_1,2\backslash cdot\; a\_2,\backslash dots,n\; \backslash cdot\; a\_n$

:$\backslash mathrm\{MG\}(a\_1,\backslash dots,a\_n)=\backslash mathrm\{MG\}(1a\_1,2a\_2,\backslash dots,na\_n)(n!)^\{-1/n\}\backslash le\; \backslash mathrm\{MA\}(1a\_1,2a\_2,\backslash dots,na\_n)(n!)^\{-1/n\}$

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

:$(n!)^\{-1/n\}\; \backslash le\; \backslash frac\{\backslash mathrm\{e\}\}\{n+1\}$ for all $n\backslash ge1.$

Therefore,

:$MG(a\_1,\backslash dots,a\_n)\; \backslash le\; \backslash frac\{\backslash mathrm\{e\}\}\{n(n+1)\}\backslash ,\; \backslash sum\_\{1\backslash le\; k\; \backslash le\; n\}\; k\; a\_k\; \backslash ,\; ,$

whence

:$\backslash sum\_\{n\backslash ge1\}MG(a\_1,\backslash dots,a\_n)\; \backslash le\backslash ,\; \backslash mathrm\{e\}\backslash ,\; \backslash sum\_\{k\backslash ge1\}\; \backslash bigg(\; \backslash sum\_\{n\backslash ge\; k\}\; \backslash frac\{1\}\{n(n+1)\}\backslash bigg)\; \backslash ,\; k\; a\_k\; =\backslash ,\; \backslash mathrm\{e\}\backslash ,\; \backslash sum\_\{k\backslash ge1\}\backslash ,\; a\_k\; \backslash ,\; ,$

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 $k=1,\backslash dots,n$. 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{{cite book |last1=Hardy |first1=G. H. |last2=Littlewood |first2=J.E. |last3=Pólya |first3=G. |title=Inequalities |date=1952 |location=Cambridge, UK |edition=Second}}{{rp|§334}}

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash left\; (\backslash frac\{a\_1+a\_2+\backslash cdots\; +a\_n\}\{n\}\backslash right\; )^p\backslash le\; \backslash left\; (\backslash frac\{p\}\{p-1\}\backslash right\; )^p\backslash sum\_\{n=1\}^\backslash infty\; a\_n^p$

for the non-negative numbers $a\_1$, $a\_2$,… and $p\; >\; 1$, replacing each $a\_n$ with $a\_n^\{1/p\}$, and letting $p\; \backslash to\; \backslash infty$.

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 $a\_i=\backslash frac\{1\}\{p\_i\}$.{{cite journal |last1=Christian Axler, Medhi Hassani |title=Carleman's Inequality over prime numbers |journal=Integers |volume=21, Article A53 |url=http://math.colgate.edu/~integers/v53/v53.pdf |access-date=13 November 2022}} They found that if $a\_i=p\_i$ one can replace $e$ with $\backslash frac\{1\}\{e\}$ in Carleman's inequality, but that if $a\_i=\backslash frac\{1\}\{p\_i\}$ then $e$ remained the best possible constant.

• {{cite book

| last = Hardy

| first = G. H. |author2=Littlewood J.E. |author3=Pólya, G.

| title = Inequalities, 2nd ed

| publisher = Cambridge University Press

| year = 1952

| pages =

| isbn = 0-521-35880-9

}}

• {{cite book

| editor-last = Rassias

| editor-first = Thermistocles M.

| title = Survey on classical inequalities

| publisher = Kluwer Academic

| year = 2000

| pages =

| isbn = 0-7923-6483-X

}}

• {{cite book

| last = Hörmander

| first = Lars

| title = The analysis of linear partial differential operators I: distribution theory and Fourier analysis, 2nd ed

| publisher = Springer

| year = 1990

| pages =

| isbn = 3-540-52343-X

}}

• {{springer|title=Carleman inequality|id=p/c020410}}

Category:Real analysis

Category:Inequalities (mathematics)