Difference between revisions of "Carleman's Inequality"
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* Steele, J. M., ''The Cauchy-Schwarz Master Class'', Cambridge University Press, ISBN 0-521-54677-X. | * Steele, J. M., ''The Cauchy-Schwarz Master Class'', Cambridge University Press, ISBN 0-521-54677-X. | ||
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Latest revision as of 16:39, 29 December 2021
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.