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The classical form of Grüss’ inequality gives an estimate of the difference between the integral of the product and the product of the integrals of two functions in C[a, b]. It was first published by G. Grüss in [7]. The aim of this article is to discuss Grüss-type inequalities in C(X), the set of continuous functions defined on a compact metric space X. We consider a functional L(f) := H(f; x), where H : C(X) → C(X) is a positive linear operator and x Î X is fixed. Generalizing a result of Acu et al. [1], a quantitative Grüss-type inequality is obtained in terms of the least concave majorant of the classical modulus of continuity. The interest is in the degree of non-multiplicativity of the functional L. Moreover, for the case X = [a, b] we improve the inequality and apply it to various known operators, in particular those of Bernstein-, convolution- and Shepard-type.

Mathematics Subject Classification (2010): 47A63, 41A25, 47B38.

Keywords: Grüss-type inequality, compact metric space, least concave majorant of the modulus of continuity, convolution-type operator, Shepard interpolation operator.