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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Gronwall inequality for weighted integrals

Authors: F. M. Wright, M. L. Klasi and D. R. Kennebeck
Journal: Proc. Amer. Math. Soc. 30 (1971), 504-510
MSC: Primary 26.45; Secondary 28.00
MathSciNet review: 0283147
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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.

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Keywords: Mean sigma integral, interior refinement integral, weighted refinement integral, Gronwall inequality, substitution for weighted integrals, product integral
Article copyright: © Copyright 1971 American Mathematical Society