Carleman’s inequality

Theorem ([1], pp. 24) For positive real numbers $\{a_{n}\}_{n=1}^{\infty}$ , Carleman’s inequality states that

\sum_{n=1}^{\infty}\big{(}a_{1}a_{2}\cdots a_{n}\big{)}^{1/n}\leq e\sum_{n=1}^% {\infty}a_{n}.

Although the constant $e$ (the natural log base) is optimal, it is possible to refine Carleman’s inequality by decreasing the weight coefficients on the right hand side [2].

References

1 L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
2 B.Q. Yuan, Refinements of Carleman’s inequality, Journal of Inequalities in Pure and Applied Mathematics, Vol. 2, Issue 2, 2001, Article 21. http://jipam.vu.edu.au/v2n2/029_00.htmlonline

Title	Carleman’s inequality
Canonical name	CarlemansInequality
Date of creation	2013-03-22 13:43:17
Last modified on	2013-03-22 13:43:17
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	5
Author	Koro (127)
Entry type	Definition
Classification	msc 26D15