Reverses of Operator Féjer's Inequalities

Abstract

Let $f$ be an operator convex 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 Gâteaux differentiable on \begin{equation*} [ A,B] :=\left\{ ( 1-t) A+tB \mid t\in [ 0,1] \right\} \,, \end{equation*} while $p:[ 0,1] \rightarrow \lbrack 0,\infty )$ is Lebesgue integrable and symmetric, namely $p\left( 1-t\right) $ $=p\left( t\right) $ for all $t\in [ 0,1] ,$ then \begin{align*} 0& \leq \int_{0}^{1}p\left( t\right) f\left( \left( 1-t\right) A+tB\right) dt-\left( \int_{0}^{1}p\left( t\right) dt\right) f\left( \frac{A+B}{2}\right) \\ & \leq \frac{1}{2}\left( \int_{0}^{1}\left\vert t-\frac{1}{2}\right\vert p\left( t\right) dt\right) \left[ \nabla f_{B}\left( B-A\right) -\nabla f_{A}\left( B-A\right) \right] \end{align*} and \begin{align*} 0& \leq \left( \int_{0}^{1}p\left( t\right) dt\right) \frac{f\left( A\right) +f\left( B\right) }{2}-\int_{0}^{1}p\left( t\right) f\left( \left( 1-t\right) A+tB\right) dt \\ & \leq \frac{1}{2}\int_{0}^{1}\left( \frac{1}{2}-\left\vert t-\frac{1}{2} \right\vert \right) p\left( t\right) dt\left[ \nabla f_{B}\left( B-A\right) -\nabla f_{A}\left( B-A\right) \right] \,. \end{align*} Two particular examples of interest are also given.