Carleman's Inequality
Carleman's Inequality states that for nonnegative real numbers , unless all the are equal to zero.
Proof
Define . Then for all positive integers , Thus Now, by AM-GM, But , so for any integer , Therefore Since for all integers , the desired inequality holds.
See also
References
- Steele, J. M., The Cauchy-Schwarz Master Class, Cambridge University Press, ISBN 0-521-54677-X.