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

DOI:
https://doi.org/10.1090/S0002-9939-1971-0283147-4

MathSciNet review:
0283147

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Abstract: The objective of this paper is to establish a Gronwall inequality for the weighted refinement integral . 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 . Helton's result contains the result here for the special case where and are nonnegative real numbers such that . 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 that is more general than the linear function considered by Helton.

**[1]**F. M. Wright and J. D. Baker,*On integration-by-parts for weighted integrals*, Proc. Amer. Math. Soc.**22**(1969), 42-52. MR**39**#7056. MR**0245750 (39:7056)****[2]**W. W. Schmaedeke and G. R. Sell,*The Gronwall inequality for modified Stieltjes integrals*, Proc. Amer. Math. Soc.**19**(1968), 1217-1222. MR**37**#6422. MR**0230864 (37:6422)****[3]**B. W. Helton,*A product integral representation for a Gronwall inequality*, Proc. Amer. Math. Soc.**23**(1969), 493-500. MR**40**#1562. MR**0248310 (40:1562)****[4]**J. V. Herod,*A Gronwall inequality for linear Stieltjes integrals*, Proc. Amer. Math. Soc.**23**(1969), 34-36. MR**40**#2802. MR**0249557 (40:2802)****[5]**F. M. Wright and J. D. Baker,*On substitution for weighted integrals*, Notices Amer. Math. Soc.**16**(1969), 206-207. Abstract #663-416.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0283147-4

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