SOME OPERATOR INEQUALITIES ON SCHUR PRODUCT AND MATRIX WEIGHTED GEOMETRIC MEAN

We complement some properties on operator inequality related to Schur product. Then we discuss the inequalities around the matrix weighted geometric mean, and give
Φ(B]tA) ≤ Φ(B)]tΦ(A);
for any positive and unital linear map Φ. Further, if mI ≤ A; B ≤ MI, then
Φ(A]tB) ≥ 4mM
(M + m)2 Φ(A)]tΦ(B):