Authors’ abstract: We give a refinement of Jensen’s inequality for \(n\)-tuples of self-adjoint operators, unital \(n\)-tuples of positive linear mappings and real valued continuous convex functions with condition on the spectra of the operators. The refined Jensen’s inequality is used to obtain a refinement of inequalities among quasi-arithmetic means under similar conditions. As an application of these results, we give a refinement of inequalities among power means.