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Specht’s ratio in the Young inequality. (English) Zbl 1021.47010

Summary: The Young operator inequality is represented for \(\lambda\in [0,1]\) as follows \[ A\nabla_{1-\lambda} B\geq A\#_{1-\lambda}B \] for positive invertible operators \(A\) and \(B\) with \(0<m\leq A\), \(B\leq M\), \(m<M\). In this note, we show the following converse inequality of the Young operator inequality on the ratio, independent of \(\lambda\): \[ S(h)A \#_{1-\lambda}B\geq A\nabla_{1-\lambda} B(\geq A\#_{1-\lambda} B), \] where the constant \[ S(h)= {h^{1\over h-1} \over e\log h^{1\over h-1}}\quad (h=\tfrac{M}{m}) \] is Specht’s ratio. Moreover, we show another converse inequality of it on the difference: \[ L(1,h)\log S(h)A\geq A\nabla_{1-\lambda} B-A\#_{1-\lambda} B(\geq 0), \] where \(L(m,M)= {M-m\over\log M-\log m}\) is the logarithmic mean.

MSC:

47A63 Linear operator inequalities
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