Inequalities involving unitarily invariant norms and operator monotone functions Original Research Article

Let short parallel·short parallel 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 tmaps toshort parallel AtXB1−tr short parallel·short parallel A1−tXBtrshort parallel is convex on [0,1] for each r>0. This convexity result interpolates the matrix Cauchy–Schwarz inequality short parallel A1/2XB1/2r short parallel2less-than-or-equals, slantshort parallel AXr short parallel·short parallel XBr short parallel due to R. Bhatia and C. Davis [Linear Algebra Appl. 223/224 (1995) 119], and also it generalizes A.W. Marshall and I. Olkinʹs [Pacific J. Math. 15 (1965) 241] result that the condition number short parallelAsshort parallel·short parallelA−sshort parallel is increasing in s>0. We prove that if f(t) is a nonnegative operator monotone function on [0,∞) and short parallel·short parallel is a normalized unitarily invariant norm, then f(short parallelXshort parallel)less-than-or-equals, slantshort parallelf(X)short parallel for every matrix X. The special case when f(t)=tr (0