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

Summary: The Young operator inequality is represented for λ[0,1] as follows

A 1-λ BA# 1-λ B

for positive invertible operators A and B with 0<mA, BM, m<M. In this note, we show the following converse inequality of the Young operator inequality on the ratio, independent of λ:

S(h)A# 1-λ BA 1-λ B(A# 1-λ B),

where the constant

S(h)=h 1 h-1 elogh 1 h-1 (h=M m)

is Specht’s ratio. Moreover, we show another converse inequality of it on the difference:

L(1,h)logS(h)AA 1-λ B-A# 1-λ B(0),

where L(m,M)=M-m logM-logm is the logarithmic mean.

47A63Operator inequalities