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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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