## On some refinement of the Cauchy–Schwarz inequality.(English)Zbl 1121.47010

This papers deals with operator means in the sense of F. Kubo and T. Ando [Math. Ann. 246, 205–224 (1979; Zbl 0412.47013)]. Denote by # the geometric mean and by $$\sigma$$ an arbitrary operator mean acting on the cone of positively semidefinite operators on a Hilbert space. Then the following generalization of the Cauchy–Schwarz inequality \begin{aligned} (A\#B)\otimes(A\#B) & \leq\frac{1}{2}\left[(A\sigma B)\otimes(A\sigma^{\bot}B)+(A\sigma^{\bot}B)\otimes(A\sigma B)\right] \\ & \leq\frac{1}{2}\left[(A\otimes B)+(B\otimes A)\right]\end{aligned} holds. Here, $$\sigma^{\bot}$$ is the dual of $$\sigma$$, defined by $$A\sigma ^{\bot}B=(B^{-1}\sigma A^{-1})^{-1}$$ for $$A>0$$, $$B>0$$. This represents a noncommutative analogue of some well-known inequalities for families of nonnegative numbers.

### MSC:

 47A63 Linear operator inequalities 26D15 Inequalities for sums, series and integrals

Zbl 0412.47013
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### References:

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