Inequalities for positive linear maps on Hermitian matrices

The aim of this work is to generalize the main inequalities in [9] as follows: Let A be a Hermitian matrix, let Phi be a normalized positive linear map, let f and g be real valued continuous functions and let F(u, v) be a real valued function matrix non-decreasing in its first variable. Real constants alpha and beta such that alpha1 less than or equal to F[Phi>(*) over bar * (f(A)),g(Phi>(*) over bar * (A))] less than or equal to beta1 are determined. If f is a concave (resp. convex) function then the determination of beta (resp. alpha) is reduced to solving a single variable maximization (resp. minimization) problem. Some applications of these results to the power function, the means and the Hadamard product are also given.