Carleman inequality

The inequality

$$\sum_{n=1}^\infty(a_1\ldots a_n)^{1/n}<e\sum_{n=1}^\infty a_n$$

for arbitrary non-negative numbers $a_n\geq0$; discovered by T. Carleman [1]. Here the constant $e$ can not be made smaller. The analogue of the Carleman inequality for integrals has the form:

$$\int\limits_0^\infty\exp\left\lbrace\frac1x\int\limits_0^x\ln f(t)dt\right\rbrace dx<e\int\limits_0^\infty f(x)dx,\quad f(x)\geq0,$$

There are also other generalizations of the Carleman inequality, [2].

References

[1]	T. Carleman, , Wissenschaft. Vorträge 5. Kongress Skandinavischen Mathematiker , Helsinki (1923) pp. 181–196
[2]	G.H. Hardy, J.E. Littlewood, G. Pólya, "Inequalities" , Cambridge Univ. Press (1934)

Comments

The inequalities are strict, except for the trivial cases $a_n=0$ for all $n$ and $f=0$ almost-everywhere.

References

[a1]	P.S. Bullen, D.S. Mitrinović, P.M. Vasić, "Means and their inequalities" , Reidel (1987)