The Gr uss inequality, as a complement of Chebyshev’s inequality, states that if f and g are integrable real functions on [a, b] and there exist real constants varphi, phi, gamma, Gamma such that varphi leq f(x) leq phi and gamma leq g(x) leq Gamma hold for all x in [a, b] , then left| frac{1}{b-a} int_a^b f(x)g(x)dx - frac{1}{(b-a)^2} int_a^b f(x)dx int_a^b g(x)dx right| leq frac{1}{4}( phi - varphi)( Gamma - gamma) ,. In this paper we present several operator Gr uss inequalities concerning positive linear maps cite{3} and Hilbert C^* -modules cite{1, 2} .

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