| Title:
|
Norm inequalities for the difference between weighted and integral means of operator differentiable functions (English) |
| Author:
|
Dragomir, Silvestru Sever |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
56 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
183-197 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f$ be a continuous function on $I$ and $A$, $B\in \mathcal{SA}_{I}\left( H\right) $, the convex set of selfadjoint operators with spectra in $I$. If $A\neq B$ and $f$, as an operator function, is Gateaux differentiable on \begin{equation*} [ A,B] :=\left\{ ( 1-t) A+tB\mid t\in \left[ 0,1\right] \right\}\,, \end{equation*} while $p\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ is Lebesgue integrable, then we have the inequalities \begin{align*} \Big\Vert \int_{0}^{1}p\left( \tau \right)& f\left( \left( 1-\tau \right) A+\tau B\right) d\tau -\int_{0}^{1}p\left( \tau \right) \,d\tau \int_{0}^{1}f\left( \left( 1-\tau \right) A+\tau B\right)\, d\tau \Big\Vert \\ & \leq \int_{0}^{1}\tau ( 1-\tau) \Big\vert \frac{\int_{\tau }^{1}p\left( s\right)\, ds}{1-\tau }-\frac{\int_{0}^{\tau }p\left( s\right)\, ds}{\tau }\Big\vert \left\Vert \nabla f_{\left( 1-\tau \right) A+\tau B}\left( B-A\right) \right\Vert \,d\tau \\ & \leq \frac{1}{4}\int_{0}^{1}\Big\vert \frac{\int_{\tau }^{1}p\left( s\right)\, ds}{1-\tau }-\frac{\int_{0}^{\tau }p\left( s\right)\, ds}{\tau } \Big\vert \left\Vert \nabla f_{\left( 1-\tau \right) A+\tau B}\left( B-A\right) \right\Vert\, d\tau\,, \end{align*} where $\nabla f$ is the Gateaux derivative of $f$. (English) |
| Keyword:
|
operator Gâteaux differentiable functions |
| Keyword:
|
integral inequalities |
| Keyword:
|
Hermite-Hadamard inequality |
| Keyword:
|
Féjer’s inequalities |
| Keyword:
|
weighted integral means |
| MSC:
|
47A63 |
| MSC:
|
47A99 |
| idZBL:
|
Zbl 07250678 |
| idMR:
|
MR4156444 |
| DOI:
|
10.5817/AM2020-3-183 |
| . |
| Date available:
|
2020-09-02T08:54:18Z |
| Last updated:
|
2020-11-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148295 |
| . |
| Reference:
|
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