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STUDIA MATHEMATICA  Issue no. 2 / 2011  
Article: 
ON GRÜSSTYPE INEQUALITIES FOR POSITIVE LINEAR OPERATORS. Authors: . 

Abstract: 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 Grüsstype 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 Grüsstype inequality is obtained in terms of the least concave majorant of the classical modulus of continuity. The interest is in the degree of nonmultiplicativity 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 Shepardtype. Mathematics Subject Classification (2010): 47A63, 41A25, 47B38. Keywords: Grüsstype inequality, compact metric space, least concave majorant of the modulus of continuity, convolutiontype operator, Shepard interpolation operator. 
