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Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators. (English) Zbl 1275.47038
For positive invertible operators \(A\), \(B\) on a Hilbert space \(\mathcal {H}\), the weighted arithmetic-geometric mean inequality holds: \(A\#_\alpha B\leq A\nabla _\alpha B\), \(\alpha \in [0,1]\). Here, \(A\nabla _\alpha B=(1-\alpha)A+\alpha B\) and \(A\#_\alpha B=A^{1/2}(A^{-1/2}BA^{-1/2})^{\alpha }A^{1/2}\). \(A\nabla _\alpha B\) is called the weighted arithmetic mean of \(A,B\) and \(A\#_\alpha B\) is called the weighted geometric mean of \(A, B\). Let \(B\) be a positive invertible operator on \(\mathcal{H}\), with \(C\) an invertible operator on \(\mathcal{H}\), \(A=C^*C\) and \((p_1,p_2) \in \mathbb{R}_+^2\). In this paper, the authors prove the inequalities: \[ 2\max \{p_1,p_2\}[A\nabla _{1/2} B-C^*(C^{*-1}BC^{-1})^{1/2}C] \] \[ \geq (p_1+p_2)[A\nabla _{\frac{p_1}{p_1+p_2}}B-C^*( C^{*-1}BC^{-1})^{\frac{p_1}{p_1+p_2}}C] \] \[ \geq 2\min \{p_1,p_2\}[A\nabla _{1/2} B-C^*(C^{*-1}BC^{-1})^{1/2}C]. \] This generalizes a similar inequality proved by F. Kittaneh and Y. Manasrah [Linear Multilinear Algebra 59, No. 9, 1031–1037 (2011; Zbl 1225.15022)]. Several related inequalities are discussed. As an application, it is shown that the Heinz mean interpolates between the arithmetic mean and the geometric mean.

47A63 Linear operator inequalities
26D15 Inequalities for sums, series and integrals
47A64 Operator means involving linear operators, shorted linear operators, etc.
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