Complementary and refined inequalities of Callebaut inequality for operators
| Autor: | Bakherad, M., Moslehian, M. S. |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | Bakherad, Mojtaba; Moslehian, Mohammad Sal. Complementary and refined inequalities of Callebaut inequality for operators. Linear Multilinear Algebra 63 (2015), no. 8, 1678--1692 |
| Druh dokumentu: | Working Paper |
| Popis: | The Callebaut inequality says that \begin{align*} \sum_{ j=1}^n \left(A_j\sharp B_j\right)\leq \left(\sum_{ j=1}^n A_j \sigma B_j\right)\sharp\left(\sum_{ j=1}^n A_j \sigma^{\bot} B_j\right)\leq\left(\sum_{ j=1}^n A_j\right)\sharp \left(\sum_{ j=1}^nB_j\right)\,, \end{align*} where $A_j, B_j\,\,(1\leq j\leq n)$ are positive invertible operators and $\sigma$ and $\sigma^\perp$ are an operator mean and its dual in the sense of Kabo and Ando, respectively. In this paper we employ the Mond--Pe\v{c}ari\'c method as well as some operator techniques to establish a complementary inequality to the above one under mild conditions. We also present some refinements of a Callebaut type inequality involving the weighted geometric mean and Hadamard products of Hilbert space operators. Comment: 15 pages, to appear in Linear Multilinear Algebra (LAMA) |
| Databáze: | arXiv |
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