Carleman’s inequality

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

$$\sum _{n=1}^{\mathrm{\infty }}{\left(a_1{a}_2\mathrm{\cdots }{a}_n\right)}^{1/n}\le e\sum _{n=1}^{\mathrm{\infty }}{a}_n.$$

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].

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