Ukrainian Mathematical Journal

, Volume 63, Issue 6, pp 843–864 | Cite as

Grüss-type and Ostrowski-type inequalities in approximation theory

  • A. M. Acu
  • H. Gonska
  • I. Raşa

We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér operator.


Approximation Theory Chebyshev Polynomial Type Inequality Interpolation Operator Positive Linear Operator 
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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • A. M. Acu
    • 1
  • H. Gonska
    • 2
  • I. Raşa
    • 3
  1. 1.Lucian Blaga University of SibiuSibiuRomania
  2. 2.University of Duisburg-EssenDuisburgGermany
  3. 3.Technical UniversityCluj-NapocaRomania

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