Carleman's Inequality

Let $\{a_i\}_{i=1}^n$ be a Set of Positive numbers. Then the Geometric Mean and Arithmetic Mean satisfy

$$\sum_{i=1}^n (a_1 a_2\cdots a_i)^{1/i}\leq {e\over n}\sum_{i=1}^n a_i.$$

Here, the constant e is the best possible, in the sense that counterexamples can be constructed for any stricter Inequality which uses a smaller constant.

See also Arithmetic Mean, e, Geometric Mean

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1094, 1979.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 249-250, 1988.