Some inequalities for weighted and integral means of operator convex functions

Abstract

Let $f$ be an operator convex function on $I$ and $A,B\mathrm{\hspace{0.17em}ϵ\hspace{0.17em}}S{A}_I\left(H\right),$ the convex set of selfadjoint operators with spectra in $I$. If $A\ne B$ and $f$, as an operator function, is Gâteaux differentiable on

$$\left[A,B\right] := \left\{\left(1-t\right)A+tB\text{|}t\in \left[0,1\right]\right\}\mathrm{,}$$

while $p : \left[0,1\right]\to [0,\infty )$ is Lebesgue integrable satisfying the condition

$$0\le {\displaystyle {\int }_0^{\tau }p}(s)ds\le {\displaystyle {\int }_0^{1}p}(s)ds\text{ for all }\tau \in [0,1]$$

and symmetric, namely $p\left(1-t\right)$ $=p\left(t\right)$ for all $tϵ\left[0,1\right]$ then

$$\begin{array}{l}-\frac{1}{{\displaystyle {\int }_0^{1}p}(\tau )d\tau }{\displaystyle {\int }_0^1\left({\displaystyle {\int }_0^{\tau }p}(s)ds\right)}(1-\tau )d\tau [\nabla f_B(B-A)-\nabla f_A(B-A)]\\ \le \frac{1}{{\displaystyle {\int }_0^{1}p}(\tau )d\tau }{\displaystyle {\int }_0^{1}p}(\tau )f((1-\tau )A+\tau B)d\tau -{\displaystyle {\int }_0^{1}f}((1-\tau )A+\tau B)d\tau \\ \le \frac{1}{{\displaystyle {\int }_0^{1}p}(\tau )d\tau }{\displaystyle {\int }_0^1\left({\displaystyle {\int }_0^{\tau }p}(s)ds\right)}(1-\tau )d\tau [\nabla f_B(B-A)-\nabla f_A(B-A)].\end{array}$$

Some particular examples of interest are also given.