Abstract
Here we present very general multivariate tight integral inequalities of Chebyshev–Grüss,Ostrowski types and of comparison of integral means. These rely on the well-known Sobolev integral representation of a function. The inequalities engage ordinary and weak partial derivatives of the involved functions.We give also applications. On the way to prove the main results we obtain important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. The exposed results are thoroughly discussed. This chapter relies on [4].
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References
G.A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2000.
G.A. Anastassiou, Probabilistic Inequalities, World Scientific, Singapore, New Jersey, 2010.
G.A. Anastassiou, Advanced Inequalities, World Scientific, Singapore, New Jersey, 2010.
G.A. Anastassiou, Multivariate Inequalities based on Sobolev Representations,Applicable Analysis, accepted 2011.
S. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer, N. York, 2008.
V. Burenkov, Sobolev spaces and domains, B.G. Teubner, Stuttgart, Leipzig, 1998.
P.L. Chebyshev, Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites, Proc. Math. Soc. Charkov, 2(1882), 93-98.
G. Grüss, Über das Maximum des absoluten Betrages von \(\left [\left ( \frac{1} {b-a}\right ){ \int \nolimits \nolimits }_{a}^{b}f\left (x\right )g\left (x\right )\mathrm{d}x\right.\) \(\left.-\left ( \frac{1} {{\left (b-a\right )}^{2}} { \int \nolimits \nolimits }_{a}^{b}f\left (x\right )\mathrm{d}x{\int \nolimits \nolimits }_{a}^{b}g\left (x\right )\mathrm{d}x\right )\right ]\), Math. Z. 39 (1935), pp. 215-226.
A. Ostrowski, Uber die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert, Comment. Math. Helv. 10(1938), 226-227.
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© 2011 George A. Anastassiou
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Anastassiou, G.A. (2011). Multivariate Integral Inequalities Deriving from Sobolev Representations. In: Inequalities Based on Sobolev Representations. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0201-5_2
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DOI: https://doi.org/10.1007/978-1-4614-0201-5_2
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