EIGENVALUE INEQUALITIES RELATED TO THE ANDO-HIAI INEQUALITY

In this paper, we show that if f is a doubly concave function on [0,8) and 0 < sA <= B <= tA for some scalars 0 < s <= t with w = t/s, then for every k = 1,2, ..., n, lambda(k)(f(A)#f(B)) <= w(1/4) +w(-1/4)/2 lambda(k)(f(A#B)), where A#B = A(1/2) ( A(-1/2) BA(-1/2))(1/2)A(1/2) is the symmetric geometric mean. As an application, we give some reverses of Ando-Hiai and Golden-Thompson type inequalities. These new reverse inequalities, improve some known results.