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A complement of the Ando–Hiai inequality. (English) Zbl 1149.47008

The authors present a complement of the generalized Ando–Hiai inequality [cf.T.Ando and F.Hiai, Linear Algebra Appl.197–198, 113–131 (1994; Zbl 0793.15011)] due to M.Fujii and E.Kamei [Linear Algebra Appl.416, No.2–3, 541–545 (2006; Zbl 1110.47011)] and find upper and lower bounds for \(A^r\sharp_{\frac{\alpha r}{(1-\alpha)s+\alpha r}}B^s\) by means of scalar multiples of \(A\sharp_{\alpha}B^\frac{rs}{(1-\alpha)s+\alpha r}\), where \(\sharp_\alpha\) denotes \(\alpha\)-geometric mean and \(0 < r \leq 1\), \(s \geq 1\).

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A63 Linear operator inequalities
47A64 Operator means involving linear operators, shorted linear operators, etc.
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References:

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