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An inequality for the Riemann-Stieltjes integral


Authors: Richard Darst and Harry Pollard
Journal: Proc. Amer. Math. Soc. 25 (1970), 912-913
MSC: Primary 26.46
DOI: https://doi.org/10.1090/S0002-9939-1970-0260945-3
MathSciNet review: 0260945
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Abstract: Let $ g$ and $ h$ be real valued and continuous on the interval $ [a,b]$, and suppose that the variation, $ V[h]$, of $ h$ on $ [a,b]$ is finite. By completely elementary methods, it is shown that $ V[h] \cdot {\sup _{_{a \leqq \alpha < \beta \leqq b}}}(g(\beta ) - g(\alpha ))$ is an upper bound for $ \int_a^b {(h - \inf h)dg} $.


References [Enhancements On Off] (What's this?)

  • [1] Tord Ganelius, Un théorème taubérien pour la transformation de Laplace, C. R. Acad. Sci. Paris 242 (1956), 719–721 (French). MR 0074579
  • [2] -, An inequality for Stieltjes integrals, Proc. Fourteenth Scandinavian Math. Congress, Copenhagen, 1964.
  • [3] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
  • [4] Rainer Wüst, Beweis eines Lemmas von Ganelius, Jber. Deutsch. Math.-Verein. 71 (1969), no. 1, 229–230. (1 plate) (German). MR 0583185

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0260945-3
Keywords: Riemann-Stieltjes integral, bounded variation, upper bound
Article copyright: © Copyright 1970 American Mathematical Society