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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.

##### MSC:
 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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