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Supplements of Hölder's Inequality

Published online by Cambridge University Press:  20 November 2018

David C. Barnes
David C. Barnes*
Affiliation:
Washington State University, Pullman, Washington
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Given vectors and (or functions f(x) and g(x)) we define the Hölder Quotient Hpq by

1

or in case of functions by

2

Here ‖·‖p and ‖·‖q are the usual Lp and Lq norms. We assume throughout that

If p and q are both greater than one then they are positive but if we allow p and q to be less than one then one of them must be positive and the other one must be negative. This may cause a problem if for example, some value ai is zero and p is negative. In this case we use the convention that and

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 3 , 01 June 1984 , pp. 405 - 420
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Hardy, G. H., Littlewood, J. E. and Pólya, G., Inequalities (Cambridge Univ. Press, London, 1964).Google Scholar
2. Mitrinović, D. S., Analytic inequalities (Springer-Verlag, New York, 1970).CrossRefGoogle Scholar