Matrix Inequalities between $f(A)σf(B)$ and $AσB$
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| Format: | Preprint |
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2023
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| _version_ | 1866929317787729920 |
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| author | Devi, Manisha Aujla, Jaspal Singh Kian, Mohsen Moslehian, Mohammad Sal |
| author_facet | Devi, Manisha Aujla, Jaspal Singh Kian, Mohsen Moslehian, Mohammad Sal |
| contents | Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $σ$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A σB)\leq \frac{f(m)}{m}(AσB)\leq f(A)σf(B)\leq \frac{f(M)}{M}(AσB)\leq f^{\prime}(M)(AσB),$$ where $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\leq\frac{f(M)}{M} |||A+B|||\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski's determinant inequality.equality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_08127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Matrix Inequalities between $f(A)σf(B)$ and $AσB$ Devi, Manisha Aujla, Jaspal Singh Kian, Mohsen Moslehian, Mohammad Sal Functional Analysis 15A60, 15A42 Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $σ$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A σB)\leq \frac{f(m)}{m}(AσB)\leq f(A)σf(B)\leq \frac{f(M)}{M}(AσB)\leq f^{\prime}(M)(AσB),$$ where $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\leq\frac{f(M)}{M} |||A+B|||\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski's determinant inequality.equality. |
| title | Matrix Inequalities between $f(A)σf(B)$ and $AσB$ |
| topic | Functional Analysis 15A60, 15A42 |
| url | https://arxiv.org/abs/2302.08127 |