Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2021-01-11 , DOI: 10.1007/s41980-020-00482-7 S. Silvestru Dragomir
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.
中文翻译：
Fréchet导数的算子凸函数的逆Jensen积分不等式
令\（f：I \ rightarrow {\ mathbb {R}} \）为类\（C ^ {1} \ left（I \ right）\）的算子凸函数。如果\（（A_ {t}）_ {t \ in T} \）是\（{\ mathcal {B}} \ left（H \ right）\）中自伴算子的有界连续字段，且频谱包含在I中定义在局部Radon测度\（\ mu \）的局部紧Hausdorff空间T上，使得\（\ int _ {T} {\ mathbf {1}} d \ mu \ left（t \ right）= \ mathbf {1，} \），那么我们得到詹森不等式的以下反向：
$$ \ begin {aligned} 0＆\ le \ int _ {T} f \ left（A_ {t} \ right）d \ mu \ left（t \ right）-f \ left（\ int _ {T} A_ { s} d \ mu \ left（s \ right）\ right）\\＆\ le \ int _ {T} Df（A_ {t}）\ left（A_ {t} \ right）d \ mu \ left（t \ right）-\ int _ {T} Df（A_ {t}）\ left（\ int _ {T} A_ {s} d \ mu \ left（s \ right）\ right）d \ mu \ left（t \ right）\ end {aligned} $$用弗雷谢特导数 \（Df（\ cdot）（\ cdot）。\）给出了Hermite-Hadamard不等式的一些应用。
京公网安备 11010802027423号