Geodesic
A unified approach to several inequalities involving functions and derivatives
Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 363-376

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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.
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.
Classification : 26D10, 26D15
Keywords: inequalities; averages of functions; quadrature 
Duoandikoetxea, Javier. A unified approach to several inequalities involving functions and derivatives. Czechoslovak Mathematical Journal, Tome 51 (2001) no. 2, pp. 363-376. http://geodesic.mathdoc.fr/item/CMJ_2001_51_2_a10/
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