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On Certain Discrete Inequalities involving Partial Sums

Published online by Cambridge University Press:  20 November 2018

Paul R. Beesack
Paul R. Beesack*
Affiliation:
Carleton University, Ottawa, Canada
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Our aim in this paper is to prove inequalities of the form

1

or

2

for all real values of the parameters α, β and all non-negative (in some cases all positive) xi. Obviously, an is finite in all cases, and we shall show that An is finite if α and α + β are both non-negative. In all cases, we obtain sharp values of the constants an, An (when finite), as well as bounds for these constants, and their behaviour as n → ∞. In case a < 0, we naturally consider only positive xi, otherwise the xi may be non-negative. Although we always write xi ≧ 0 in the following, this should be read as xi > 0 in case α < 0; similar remarks apply to the parameter t introduced below.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 222 - 234
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Lee, C.-M., On a discrete analogue of inequalities of Opial and Yang, Can. Math. Bull. 11 (1968), 73–77.Google Scholar
2. Opial, Z., Sur une inégalité, Ann. Polon. Math. 8 (1960), 29–32.Google Scholar
3. Wong, J. S. W., A discrete analogue ofOpiaVs inequality, Can. Math. Bull. 10 (1967), 115–118.Google Scholar
4. Yang, G. S., On a certain result of Z. Opial, Proc. Japan Acad. 1+2 (1966), 78–83.Google Scholar