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Some families of operator norm inequalities. (English) Zbl 1476.15033

Summary: We consider the function \(f_{\alpha, \beta}(t) = t^{\gamma(\alpha, \beta)} \prod_{i = 1}^n \frac{b_i(t^{a_i} - 1)}{a_i(t^{b_i} - 1)}\) on the interval \((0, \infty)\), where \(\alpha = (a_1, a_2, \ldots, a_n)\), \(\beta = (b_1, b_2, \ldots, b_n) \in \mathbb{R}^n\) and \(\gamma(\alpha, \beta) = (1 - \sum_{i = 1}^n(a_i - b_i)) / 2\). In [Indiana Univ. Math. J. 48, No. 3, 899–936 (1999; Zbl 0934.15023)] F. Hiai and H. Kosaki define the relation using positive definiteness for functions \(f\) and \(g\) with some suitable conditions and they have proved this relation implies the operator norm inequality associated with functions \(f\) and \(g\). In this paper, we give some conditions for \(\alpha^\prime, \beta^\prime \in \mathbb{R}^m\) to hold the relation \(f_{\alpha, \beta}(t) \preceq f_{\alpha^\prime, \beta^\prime}(t)\).

MSC:

15A45 Miscellaneous inequalities involving matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47A53 (Semi-) Fredholm operators; index theories
47A64 Operator means involving linear operators, shorted linear operators, etc.
15A42 Inequalities involving eigenvalues and eigenvectors

Citations:

Zbl 0934.15023
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References:

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