Klein's Trace Inequality and Superquadratic Trace Functions

M. Kian, M. Alomari
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引用次数: 1

Abstract

We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$ for all positive matrices $A,B$. It gives in particular improvement of Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.
克莱因迹不等式与超二次迹函数
我们证明了如果$f$是一个非负的超二次函数,那么$A\mapsto\mathrm{Tr}f(A)$在矩阵代数上是一个超二次函数。特别地,\begin{align*} \tr f\left( {\frac{{A + B}}{2}} \right) +\tr f\left(\left| {\frac{{A - B}}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*}适用于所有正矩阵$A,B$。此外,对于所有正矩阵$A,B$,我们给出了超二次函数为$$ \mathrm{Tr}[f(A)-f(B)-(A-B)f'(B)]\geq \mathrm{Tr}[f(|A-B|)] $$的Klein不等式。特别给出了非负凸函数的Klein不等式的改进。因此,给出了超二次函数的Jensen迹不等式的一些变体。
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