On a trace inequality Original Research Article

Let A and B be positive operators and p, α, s greater-or-equal, slanted 0. Assume either (1) A greater-or-equal, slanted B and β greater-or-equal, slanted max−½(p + 2α), − ½(1 + 2α), or (2) A and B are invertible with log A greater-or-equal, slanted log B and β greater-or-equal, slanted −α. Then, for any continuous increasing function ƒ on real+ with ƒ(0) = 0, the trace inequality Tr ƒ(Aβ(AαBpAα)sAβ) less-than-or-equals, slant Tr ƒ(A(p+2α)s+2β) holds. This generalizes both a trace inequality due to Kosaki and one due to Furuta.