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Generalization of Hölder's and Minkowski's inequalities

Published online by Cambridge University Press:  24 October 2008

D. E. Daykin  and
C. J. Eliezer
D. E. Daykin
Affiliation:
University of Malaya, Kuala Lumpur
C. J. Eliezer
Affiliation:
University of Malaya, Kuala Lumpur
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Extract

In a recent paper (1) the authors considered some generalizations of Cauchy's inequality. The method of approach was by the construction of certain convex functions (where by a convex function we mean a function f(x) satisfying

for every pair of unequal values x1 and x2). For example, it was shown that if (a), (b) are the sets of non-negative real numbers a1, …,an; bl, …, bn, the function

which had been used earlier by Callebaut(2), is a convex function with minimum at x = 0, except if the sets (a) and (b) are proportional, in which case the function is a constant. This function takes the value (σab)2 when x = 0 and (σa2) (Σb2) when x = 1, and the property of convexity gives Cauchy's inequality.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 4 , October 1968 , pp. 1023 - 1027
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Eliezer, C. J. and Daykin, D. E.Generalizations and applications of Cauchy-Schwartz inequality. Quart. J. Math. Oxford Ser. 40 (1967), 247–250.Google Scholar
(2)Callebaut, D. K.Generalization of the Cauchy–Schwartz inequality. J. Math. Anal. Appl. 12 (1965), 491–494.CrossRefGoogle Scholar
(3)Hardy, G. H., Littlewood, J. E. and Polya, G.Inequalities (Cambridge University Press, 1952).Google Scholar