Multivariate Integral Inequalities Deriving from Sobolev Representations

  • George A. AnastassiouEmail author
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


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].


Open Ball Integral Inequality Representation Proof Important Estimate Weak Partial Derivative 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G.A. Anastassiou, Quantitative Approximations, Chapman & Hall/CRC, Boca Raton, New York, 2000.CrossRefGoogle Scholar
  2. 2.
    G.A. Anastassiou, Probabilistic Inequalities, World Scientific, Singapore, New Jersey, 2010.zbMATHGoogle Scholar
  3. 3.
    G.A. Anastassiou, Advanced Inequalities, World Scientific, Singapore, New Jersey, 2010.CrossRefGoogle Scholar
  4. 4.
    G.A. Anastassiou, Multivariate Inequalities based on Sobolev Representations,Applicable Analysis, accepted 2011.Google Scholar
  5. 5.
    S. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer, N. York, 2008.Google Scholar
  6. 6.
    V. Burenkov, Sobolev spaces and domains, B.G. Teubner, Stuttgart, Leipzig, 1998.Google Scholar
  7. 7.
    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.Google Scholar
  8. 8.
    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.Google Scholar
  9. 9.
    A. Ostrowski, Uber die Absolutabweichung einer differentiabaren Funcktion von ihrem Integralmittelwert, Comment. Math. Helv. 10(1938), 226-227.CrossRefMathSciNetGoogle Scholar

Copyright information

© George A. Anastassiou 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations