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inequalities; averages of functions; quadrature
There are many inequalities measuring the deviation of the average of a function over an interval from a linear combination of values of the function and some of its derivatives. A general setting is given from which the desired inequalities are obtained using Hölder’s inequality. Moreover, sharpness of the constants is usually easy to prove by studying the equality cases of Hölder’s inequality. Comparison of averages, extension to weighted integrals and $n$-dimensional results are also given.
[1] G. A. Anastassiou: Ostrowski type inequalities. Proc. Amer. Math. Soc. 123 (1995), 3775–3781. DOI 10.1090/S0002-9939-1995-1283537-3 | MR 1283537 | Zbl 0860.26009
[2] P. J. Davis and P. Rabinowitz: Methods of Numerical Integration. Academic Press, New York, 1975. MR 0448814
[3] A. M. Fink: Bounds on the deviation of a function from its averages. Czechoslovak Math. J. 42 (1992), 289–310. MR 1179500 | Zbl 0780.26011
[4] D. S. Mitrinović, J. E. Pečarić and A. M. Fink: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991. MR 1190927
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