Reverse Jensen Integral Inequalities for Operator Convex Functions in Terms of Fréchet Derivative

Abstract

Let $f:I\rightarrow {\mathbb {R}}$ be an operator convex function of class $C^{1}\left( I\right)$. If $(A_{t})_{t\in T}$ is a bounded continuous field of selfadjoint operators in ${\mathcal {B}}\left( H\right)$ with spectra contained in I defined on a locally compact Hausdorff space T with a bounded Radon measure $\mu$, such that $\int _{T}{\mathbf {1}}d\mu \left( t\right) =\mathbf {1,}$ then we obtain among others the following reverse of Jensen’s inequality:

$$\begin{aligned} 0&\le \int _{T}f\left( A_{t}\right) d\mu \left( t\right) -f\left( \int _{T}A_{s}d\mu \left( s\right) \right) \\&\le \int _{T}Df(A_{t})\left( A_{t}\right) d\mu \left( t\right) -\int _{T}Df(A_{t})\left( \int _{T}A_{s}d\mu \left( s\right) \right) d\mu \left( t\right) \end{aligned}$$

in terms of the Fréchet derivative $Df(\cdot )(\cdot ).$ Some applications for the Hermite–Hadamard inequalities are also given.