Interpolating inequalities for functions of positive semidefinite matrices

Abstract

Let A, B be positive semidefinite n×n matrices, and let α∈(0,1). We show that if f is an increasing submultiplicative function on [0,∞) with f(0)=0 such that f(t) and f2(t1/2) are convex, then |||f(AB)|||2≤f4(1(4α(1−α))1/4)(|||(αf(A)+(1−α)f(B))2|||×|||((1−α)f(A)+αf(B))2|||) for every unitarily invariant norm. Moreover, if α∈[0,1] and X is an n×n matrix with X≠0, then |||f(AXB)|||2≤f(‖X‖)‖X‖|||αf2(A)X+(1−α)Xf2(B)||||||(1−α)f2(A)X+αXf2(B)||| for every unitarily invariant norm. These inequalities present generalizations of recent results of Zou and Jiang and of Audenaert.