Reverses and variations of Heinz inequality

Let A, B be positive definite n x n matrices. We present several reverse Heinz-type inequalities, in particular parallel to AX + XB parallel to(2)(2) + 2(nu - 1) parallel to AX - XB parallel to(2)(2) <= parallel to A(nu)XB(1-nu) + A(1-nu)XB(nu)parallel to(2)(2), where X is an arbitrary n x n matrix, parallel to center dot parallel to(2) is Hilbert-Schmidt norm and nu > 1. We also establish a Heinz-type inequality involving Hadamard product of the form 2 vertical bar parallel to A(1/2) o B-1/2 vertical bar parallel to <= vertical bar parallel to A(s) o B1-t + A(1-s) o B-t vertical bar parallel to <= max{vertical bar parallel to(A + B) o I vertical bar parallel to, vertical bar parallel to(A o B) + I vertical bar parallel to}, in which s, t is an element of [ 0, 1] and vertical bar parallel to center dot vertical bar parallel to is a unitarily invariant norm.