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On an extension of Copson's inequality for infinite series

Published online by Cambridge University Press:  14 November 2011

B. M. Brown  and
W. D. Evans
B. M. Brown
Affiliation:
Department of Computing Mathematics, University of Wales College of Cardiff, Senghennydd Road, Cardiff CF2 4YN, Wales, U.K
W. D. Evans
Affiliation:
School of Mathematics, University of Wales College of Cardiff, Senghennydd Road, Cardiff CF2 4AG, Wales, U.K
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Synopsis

In 1979 Copson proved the following analogue of the Hardy-Littlewood inequality: if is a sequence of real numbers such that are convergent, where Δan = an+1 – an and Δ2an = Δ(Δan), then is convergent and the constant 4 being best possible. Equality occurs if and only if an = 0 for all n. In this paper we give a result that extends Copson's result to inequalities of the form

where Mxn =–Δ(pn_l Δxn_l)+qnxn (n = 0, 1, …). The validity of such an inequality and the best possible value of the constant K are determined in terms of the analogue of the Titchmarsh-Weyl m-function for the difference equation Mxn = λwnxn (n = 0, 1, …).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 121 , Issue 1-2 , 1992 , pp. 169 - 183
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Atkinson, F. V.. Discrete and Continuous Boundary Problems (New York: Academic Press, 1964).Google Scholar
2Bennewitz, C.. Spectral theory for pairs of differential operators. Ark. Mat. 15 (1977), 33–61.CrossRefGoogle Scholar
3Chaudhuri, J. and Everitt, W. N.. On the spectrum of ordinary second-order differential operators. Proc. Roy. Soc. Edinburgh Sect. A 68 (1968), 95–119.Google Scholar
4Copson, E. T.. Two Series Inequalities. Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), 109–114.CrossRefGoogle Scholar
5Evans, W. D. and Everitt, W. N.. A Return to the Hardy Littlewood Integral Inequality. Proc. Roy. Soc. London Ser. A 380 (1982), 447–486.Google Scholar
6Evans, W. D. and Zettl, A.. Norm inequalities involving derivatives. Proc. Roy. Soc. Edinburgh Sect. A 82 (1978), 51–70.CrossRefGoogle Scholar
7Everitt, W. N.. On an extension to an integro-differential inequality of Hardy, Littlewood and Pólya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1971/1972), 295–333.Google Scholar
8Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Cambridge: Cambridge University Press, 1934).Google Scholar
9Hinton, D. B. and Lewis, R. T.. Spectral analysis of second-order difference equations. J. Math. Anal. Appl. 63 (1978), 421–438.CrossRefGoogle Scholar
10Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations, I, 2nd edn (Oxford: Oxford University Press, 1962).CrossRefGoogle Scholar