# 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
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