On the convexity of functions

Omar Hirzallah, Ata Abu-As'ad


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


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


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