Inequalities involving unitarily invariant norms and operator monotone functions

Let parallel to . parallel to be a unitarily invariant norm on matrices. For matrices A, B, X with A, B positive semidefinite and X arbitrary, we prove that the function t --> parallel to \A(t) x B1-t\(r) parallel to . parallel to \A(1-t) x B-t\(r) parallel to is convex on [0, 1] for each r > 0. This convexity result interpolates the matrix Cauchy-Schwarz inequality parallel to \A(1/2) x B-1/2\(r) parallel to(2) less than or equal to parallel to \AX\(r) parallel to . parallel to \XB\(r) parallel to due to R. Bhatia and C. Davis [Linear Algebra Appl. 223/224 (1995) 119], and also it generalizes A.W. Marshall and I. OLkin's [Pacific J. Math. 15 (1965) 241] result that the condition number parallel toA(s)parallel to . parallel toA(-s)parallel to is increasing in s > 0. We prove that if f (t) is a nonnegative operator monotone function on [0, infinity) and parallel to parallel to is a normalized unitarily invariant norm, then f (parallel to X parallel to) less than or equal to parallel to f(\X\) parallel to for every matrix X. The special case when f (t) = t(r) (0 < r less than or equal to 1) is used to consider the monotonicity of p --> parallel to A(p) + B-p parallel to (1/p) as well as p --> parallel to (A(p)+B-p) (1/p) parallel to. Furthermore, we obtain some norm inequalities of Hblder and Minkowski types related to the expression parallel to \A\(p)+\B\(p) parallel to (1/p). For example, comparisons are made between parallel to C*A + D*B parallel to and parallel to \A\(p)+\B\(p). parallel to \C\(q)+\D\(q) parallel to(1/q), where p(-1)+q(-1) = 1. (C) 2002 Elsevier Science Inc. All rights reserved.