Extensions of interpolation between the arithmetic-geometric mean inequality for matrices

In this paper, we present some extensions of interpolation between the arithmetic- geometric means inequality. Among other inequalities, it is shown that if A, B, X are n x n matrices, then parallel to AXB*parallel to(2) <= parallel to f(1) (A*A)Xg(1) (B* B) parallel to parallel to f(2) (A*A) Xg(2) (B*B) parallel to, where f(1), f(2), g(1), g(2) are non-negative continuous functions such that f(1)(t)f(2)( t) = t and g(1)(t)g(2)(t) = t (t >= 0). We also obtain the inequality |||AB*|||(2) <= |||p(A* A)(m/p) + (1 - p)(B* B)(s/1-p) ||| ||| (1 - p)(A* A)(n/1-p) + p(B* B)(t/p) |||, in which m, n, s, t are real numbers such that m+ n = s + t = 1, ||| . ||| is an arbitrary unitarily invariant norm and p is an element of[0, 1].