On Gruss-type inequalities for positive linear operators

Abstract

The classical form of Gruss’ 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. Gruss in [7]. The aim of this article is to discuss Gruss-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 ∈ X is fixed. Generalizing a result of Acu et al. [1], a quantitative Gruss-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-, convolutionand Shepard-type. Mathematics Subject Classification (2010): 47A63, 41A25, 47B38.