SINGULAR VALUE INEQUALITIES RELATED TO THE AUDENAERT-ZHAN INEQUALITY

In this paper, we refine the Heinz mean inequality for singular values and give some generalizations of Audenaert-Zhan inequality for singular values and Zhan's conjecture for the case of negative t. Among others, we show that if A, B is an element of M-n are positive semidefinite and f, g are real valued continuous functions on [0, infinity) such that g is monotone and f (g(-1)(root t))(2) is operator monotone on [0, infinity), then s(j)(f(A)(g(A)(2) + g(B)(2))f (B)) <= 1/2s(j)(f(A)(2)g(A)(2) + f(B)(2)g(B)(2)) for j = 1,..., n, where s(j) are the singular values in decreasing order.