### Abstract

A weighted version of the geometric mean of k (≥̧ 3) positive invertible operators is given. For operators A_{1},... , A_{k} and for nonnegative numbers α1,... , αk such that we define weighted geometric means of two types, the first type by a direct construction through symmetrization procedure, and the second type by an indirect construction through the non-weighted (or uniformly weighted) geometric mean. Both of them reduce to A_{1}^{α} A^{α}_{k}^{k} if A_{1},... , A_{k} commute with each other. The first type does not have the property of permutation invariance, but satisfies a weaker one with respect to permutation invariance. The second type has the property of permutation invariance. We also show a reverse inequality for the arithmetic-geometric mean inequality of the weighted version.

Original language | English |
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Pages (from-to) | 213-228 |

Number of pages | 16 |

Journal | Kyungpook Mathematical Journal |

Volume | 50 |

Issue number | 2 |

Publication status | Published - Jul 2010 |

Externally published | Yes |

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### Keywords

- Arithmetic-geometric mean inequality
- Positive operator
- Reverse inequality
- Weighted geometric mean

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Kyungpook Mathematical Journal*,

*50*(2), 213-228.