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Some cyclic and other inequalities. III

Published online by Cambridge University Press:  24 October 2008

P. H. Diananda
P. H. Diananda
Affiliation:
Department of Mathematics, University of Singapore
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Extract

1. Introduction. In a recent paper (2) Daykin has made the Conjecture. If t is real, then

where

He has proved (1, 2) that (i) Φ(t) is a convex function of t, (ii) Φ(t)≥ n if t ≥ 0 or 2, and (iii) Φ(t) takes values arbitrarily close to ½n if n is even, x2 = x4 = … = xn = 1, and xl = x3 = … = xn-1 and t have suitable small positive values. It is known (3) that (1) is true if t = 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 73 , Issue 1 , January 1973 , pp. 69 - 71
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Daykin, D. E.Inequalities of a cyclic nature. J. London Math. Soc. (2) 3 (1971), 453–462.CrossRefGoogle Scholar
(2)Dayxin, D. E.Inequalities for certain cyclic sums. Proc. Edinburgh Math. Soc. (2) 17 (1971), 257–262.Google Scholar
(3)Diananda, P. H.Some cyclic and other inequalities. Proc. Cambridge Philos. Soc. 58 (1962), 425–427.CrossRefGoogle Scholar
(4)Diananda, P. H. Some cyclic and other inequalities.II. Proc. Cambridge Philos. Soc. 58 (1962), 703–705.CrossRefGoogle Scholar