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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 / D H.f I x/; where H W C OEa; b ! C OEa; b is a positive linear operator and x 2 OEa; b is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the HermiteFejér 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.

We study the degree of approximation of the iterated Bernstein operators to the members (T (t)) t=0 , of their limiting semigroup. This yields a full quantitative version of an earlier convergence result by Karlin and Ziegler.

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