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].
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 |