In this paper we study the utility of the functional Pompeiu-Chebyshev in some inequalities. Some results obtained by Alomari will be generalized regarding inequalities with Pompeiu-Chebyshev type functionals, in which linear and positive functionals intervene. We investigate some new inequalities of Grüss type using Pompeiu's mean value theorem. Improvement of known inequalities is also given.
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