Norm inequalities for the difference between weighted and integral means of operator differentiable functions

Dragomir, Silvestru Sever. "Norm inequalities for the difference between weighted and integral means of operator differentiable functions." Archivum Mathematicum 056.3 (2020): 183-197. <http://eudml.org/doc/297053>.

@article{Dragomir2020,
abstract = {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 $A\ne B$ and $f$, as an operator function, is Gateaux differentiable on \begin\{equation*\} [ A,B] :=\left\lbrace ( 1-t) A+tB\mid t\in \left[ 0,1\right] \right\rbrace \,, \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 \\ & \le \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 \\ & \le \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 $f$.},
author = {Dragomir, Silvestru Sever},
journal = {Archivum Mathematicum},
keywords = {operator Gâteaux differentiable functions; integral inequalities; Hermite-Hadamard inequality; Féjer’s inequalities; weighted integral means},
language = {eng},
number = {3},
pages = {183-197},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Norm inequalities for the difference between weighted and integral means of operator differentiable functions},
url = {http://eudml.org/doc/297053},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Dragomir, Silvestru Sever
TI - Norm inequalities for the difference between weighted and integral means of operator differentiable functions
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 3
SP - 183
EP - 197
AB - 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 $A\ne B$ and $f$, as an operator function, is Gateaux differentiable on \begin{equation*} [ A,B] :=\left\lbrace ( 1-t) A+tB\mid t\in \left[ 0,1\right] \right\rbrace \,, \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 \\ & \le \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 \\ & \le \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 $f$.
LA - eng
KW - operator Gâteaux differentiable functions; integral inequalities; Hermite-Hadamard inequality; Féjer’s inequalities; weighted integral means
UR - http://eudml.org/doc/297053
ER -