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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 Φ 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 α and β such that αI F [Φ (f (A)) , g (Φ(A))] βI are determined. If f is a concave (resp. convex) function then the determination of β (resp. α ) 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. Mathematics subject classification (1991): 47A30, 47A63.