Grüss-type and Ostrowski-type inequalities in approximation theory - Acu, A.-M.; Gonska, H.; Ra¸sa, I. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
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] → C[a,b] is a positive linear operator and x [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.