Carleman inequality

The inequality

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

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 for all and almost-everywhere.

References

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