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A Remark on Completely Monotonic Sequences, with an Application to Summability

Published online by Cambridge University Press:  20 November 2018

Lee Lorch  and
Leo Moser
Lee Lorch
Affiliation:
University of Alberta
Leo Moser
Affiliation:
University of Alberta
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A sequence of non-negative numbers, μ0 μ1 … μn, is called completely monotonic [5, p. 108] if (-1)n Δn μk ≧ 0 for n, k = 0, 1, 2, …. Such sequences occur in many connexions, such as the Hausdorff moment problem and Hausdorff summability [1, Chapter XI, 5, Chapters III and IV].

It is natural to inquire as to the circumstances under which the inequality "≧" above can be strengthened to ">". As it happens, this can be done always, except for sequences all of whose terms past the first are identical. The formal statement follows. (In it, as above, Δnμk is the n-th forward difference, i. e. Δ0μk = μk; Δnμk = Δn-1 μk+1 - Δn-1μk.)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 2 , May 1963 , pp. 171 - 173
Copyright
Copyright © Canadian Mathematical Society 1963

References

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3. Lee, Lorch and Peter, Szego, Higher monotonicity properties of certain Sturm-Liouville functions, Acta Math., (to appear).Google Scholar
4. Rogosinski, W. W., On Hausdorff's methods of summability, Proc. Cambridge Philos. Soc., vol. 38 (1942), pp. 166-192.Google Scholar
5. Widder, D.V., The Laplace Transform, Princeton, 1941.Google Scholar