A note on some inequalities for positive linear maps

Abstract

We improve and generalize some operator inequalities for positive linearmaps. It is shown, among other inequalities, that if 0 < m <= B <= m < M <= A <= M or 0 < m <= A <= m < M <= B <= M, then for each 2 <= p < 8 and.. 0, 1 , p A..B <=.. K h 4 2p -1 Kr h .. p p A#.B , and p A..B <=.. K h 4 2p -1 Kr h .. p A #. B p, where r <= min {., 1 -.}, h <= Mm and h <= M m . We also obtain an improvement of operator Polya-Szego inequality.