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Inequalities related to Hardy's and Heinig's

Published online by Cambridge University Press:  24 October 2008

James A. Cochran  and
Cheng-Shyong Lee
James A. Cochran
Affiliation:
Department of Mathematics, Washington State University, Pullman, WA 99164, U.S.A.
Cheng-Shyong Lee
Affiliation:
Department of Mathematics, Washington State University, Pullman, WA 99164, U.S.A.
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Extract

In a 1975 paper [8], Heinig established the following three inequalities:

where A = p/(p + s − λ) with p, s, λ real numbers satisfying p + s > λ,p > 0;

where B = p/(2p + sp − λ −1) with p, s, λ real numbers satisfying 2p +sp > λ, + 1, p > 0;

where is a sequence of nonnegative real numbers,

and C = p[l + l/(p + s−λ)] with p, s, λ real numbers satisfying s > 0, p ≥ 1, and p +s > λ 0.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 1 , July 1984 , pp. 1 - 7
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

[1] Abramowitz, M. and Stegun, I. A.. Handbook of Mathematical Functions. Applied Mathematics, series 55 (National Bureau of Standards, Washington, D.C., 1964).Google Scholar
[2] Carleman, T.. Sur les fonctions quasi-analytiques, Conférences faites au cinquiéme congrés des mathématiciens Scandinaves, Helsingfors (1923), pp. 181–196.Google Scholar
[3] Dunford, N. and Schwartz, J. T.. Linear Operators, part I (John Wiley, New York, 1958).Google Scholar
[4] Hardy, G. H.. Notes on some points in the integral calculus (LX). Messenger of Math. 54 (1925), 150–156.Google Scholar
[5] Hardy, G. H.. Notes on some points in the integral calculus (LXIV). Messenger of Math. 57 (1928), 12–16.Google Scholar
[6] Hardy, G. H. and Littlewood, J. E.. Notes on the theory of series (XII): on certain inequalities connected with the calculus of variations. J. London Math. Soc. 5 (1930), 34–39.CrossRefGoogle Scholar
[7] Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities, 2nd ed. (Cambridge University Press, 1964).Google Scholar
[8] Heinig, H. P.. Some extensions of Hardy's inequality. SIAM J. Math. Anal. 6 (1975), 698–713.CrossRefGoogle Scholar
[9] Hewitt, E. and Stromberg, K.. Real and Abstract Analysis (Springer-Verlag, New York, 1965).Google Scholar
[10] Knopp, K.. Über Reihen mit positiven Gliedern. London Math. Soc. 3 (1928), 205–211.CrossRefGoogle Scholar
[11] Mitrinović, D. S.. Analytic Inequalities (Springer-Verlag, New York, 1970).CrossRefGoogle Scholar