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

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 $f$