Jensen and Minkowski inequalities for operator means and anti-norms

Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means such as some p-means and some Kubo-Ando means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q∈(−∞,1] and the Minkowski functional det1/nA.