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Let f (t) be a nonnegative concave function on 0 ≤ t < ∞ with f (0) = 0, and let X, Y be n×n matrices. Then it is known that f where · 1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.doi:10.1090/s0002-9939-05-08116-5 fatcat:ck3eeincnncmhe2lp4yb6zpdje