The Gronwall inequality for weighted integrals
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- by F. M. Wright, M. L. Klasi and D. R. Kennebeck
- Proc. Amer. Math. Soc. 30 (1971), 504-510
- DOI: https://doi.org/10.1090/S0002-9939-1971-0283147-4
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Abstract:
The objective of this paper is to establish a Gronwall inequality for the weighted refinement integral $[F,({w_1},{w_2},{w_3})]\smallint _a^bf(t)dg(t)$ . This result generalizes a recent result by W. W. Schmaedeke and G. R. Sell for the mean sigma integral and the interior refinement integral. The proof in this paper is based on the one given by Schmaedeke and Sell but is shorter and simpler. B. W. Helton has established a product integral representation for a Gronwall inequality for the refinement integral $(LR)\smallint _a^b(fH + fG)$. Helton’s result contains the result here for the special case where ${w_1}$ and ${w_3}$ are nonnegative real numbers such that ${w_1} + {w_3} = 1$. The ideas used here are considerably less complicated than those used by Helton. J. V. Herod has established a Gronwall inequality for linear Stieltjes integrals working with a linear function $J[f]$ that is more general than the linear function $J[f] = (LR)\smallint _a^b(fH + fG)$ considered by Helton.References
- Fred M. Wright and James D. Baker, On integration-by-parts for weighted integrals, Proc. Amer. Math. Soc. 22 (1969), 42–52. MR 245750, DOI 10.1090/S0002-9939-1969-0245750-8
- Wayne W. Schmaedeke and George R. Sell, The Gronwall inequality for modified Stieltjes integrals, Proc. Amer. Math. Soc. 19 (1968), 1217–1222. MR 230864, DOI 10.1090/S0002-9939-1968-0230864-8
- Burrell W. Helton, A product integral representation for a Gronwall inequality, Proc. Amer. Math. Soc. 23 (1969), 493–500. MR 248310, DOI 10.1090/S0002-9939-1969-0248310-8
- J. V. Herod, A Gronwall inequality for linear Stieltjes integrals, Proc. Amer. Math. Soc. 23 (1969), 34–36. MR 249557, DOI 10.1090/S0002-9939-1969-0249557-7 F. M. Wright and J. D. Baker, On substitution for weighted integrals, Notices Amer. Math. Soc. 16 (1969), 206-207. Abstract #663-416.
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 30 (1971), 504-510
- MSC: Primary 26.45; Secondary 28.00
- DOI: https://doi.org/10.1090/S0002-9939-1971-0283147-4
- MathSciNet review: 0283147