On the Convexity of Functions

Abstract

Let A,B, and X be bounded linear operators on a separable Hilbert space such that A,B are positive, X≥γI, for some positive real number γ, and α∈[0,1]. Among other results, it is shown that if f(t) is an increasing function on [0,∞) with f(0)=0 such that f(√t) is convex, then

γ|||f(αA+(1-α)B)+f(β|A-B|)|||≤|||αf(A)X+(1-α)Xf(B)|||

for every unitarily invariant norm, where β=min(α,1-α). Applications of our results are given.