## Abstract

Let $f$ be an operator convex function on $I$ and $A,B\mathrm{\hspace{0.17em}\u03f5\hspace{0.17em}}S{A}_{I}\left(H\right),$ 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 $p:\left[0,1\right]\to [0,\text{}\infty )$ is Lebesgue integrable satisfying the condition

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

and symmetric, namely $p\left(1-t\right)$ $=p\left(t\right)$ for all $t\u03f5\text{}\left[0,\text{}1\right]$ 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.

## Acknowledgement

The author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper.

## Citation

Silvestru Sever Dragomir. "Some inequalities for weighted and integral means of operator convex functions." SUT J. Math. 56 (2) 109 - 127, December 2020. https://doi.org/10.55937/sut/1609966712

## Information