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**Refinements and reverses of means inequalities for Hilbert space operators.**
*(English)*
Zbl 1276.47021

This paper discusses improvements of a series of inequalities for Hilbert space operators.

The authors define a functional that measures the difference between the classical arithmetic and geometric means and also deduce some significant scalar inequalities. Section 3 provides, under certain conditions, improvements of the series of inequalities and establishes the lower bound for the difference between the arithmetic and geometric operator means. As an application, in Section 4, the authors establish an improved variant of an inequality concerning the Heinz operator mean. In the final section, the authors present some eigenvalue inequalities for differences of certain operator means.

The authors define a functional that measures the difference between the classical arithmetic and geometric means and also deduce some significant scalar inequalities. Section 3 provides, under certain conditions, improvements of the series of inequalities and establishes the lower bound for the difference between the arithmetic and geometric operator means. As an application, in Section 4, the authors establish an improved variant of an inequality concerning the Heinz operator mean. In the final section, the authors present some eigenvalue inequalities for differences of certain operator means.

Reviewer: V. Lokesha (Bangalore)

### MSC:

47A63 | Linear operator inequalities |

47A10 | Spectrum, resolvent |

47B06 | Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators |

47B07 | Linear operators defined by compactness properties |

47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |

26D20 | Other analytical inequalities |

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\textit{F. Kittaneh} et al., Banach J. Math. Anal. 7, No. 2, 15--29 (2013; Zbl 1276.47021)

### References:

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