Address:
Mathematics, College of Engineering and Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg 2050, South Africa
E-mail:
sever.dragomir@vu.edu.au
http://rgmia.org/dragomir
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 Gâteaux differentiable on \begin{equation*} [ A,B] :=\left\rbrace ( 1-t) A+tB\mid t\in \left[ 0,1\right] \right\lbrace \,, \end{equation*} while $p\colon \left[ 0,1\right] \rightarrow \mathbb{R}$ is Lebesgue integrable, then we have the inequalities \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 \,, where $\nabla f$ is the Gâteaux derivative of $f$.
AMSclassification: primary 47A63; secondary 47A99.
Keywords: operator Gâteaux differentiable functions, integral inequalities, Hermite-Hadamard inequality, Féjer’s inequalities, weighted integral means.
DOI: 10.5817/AM2020-3-183