Grüss-type and Ostrowski-type inequalities in approximation theory
We discuss the Grass inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain a Grass inequality for the functional $L(f) = H(f; x)$, where $H: C[a,b] \rightarrow C[a,b]$ is a positive linear operator and $x \in [a,b]$ is fixed. We apply this inequality in the case of known operators, for example, the Bernstein, Hermite-Fejer operator the interpolation operator, convolution-type operators. Moreover, we derive Grass-type inequalities using Cauchy's mean value theorem, thus generalizing results of Cebysev and Ostrowski. A Grass 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 Grass' inequality. In an appendix, we prove a new result concerning the absolute first-order moments of the classical Hermite-Fejer operator.
English version (Springer): Ukrainian Mathematical Journal 63 (2011), no. 6, pp 843-864.
Citation Example: Acu A. M., Gonska H., Ra¸sa I. Grüss-type and Ostrowski-type inequalities in approximation theory // Ukr. Mat. Zh. - 2011. - 63, № 6. - pp. 723-740.