论文标题
克莱因的痕量不平等和超界面痕量函数
Klein's trace inequality and superquadratic trace functions
论文作者
论文摘要
我们表明，如果$ f $是一种非负超偏二次函数，则$ a \ mapsto \ mathrm {tr} f（a）$是矩阵代数上的超偏函数。尤其， \ begin {align*} \ tr f \ left（{\ frac {a + b} {2}} \ right） + \ tr f \ left（\ left | {\ frac {\ frac {a- b} {2}} {2}} \ right | \ right | \ right | right） \ right）}}}} {2} \ end {align*} 持有所有正矩阵$ a，b $。 此外，我们提出了克莱因（Klein \ Mathrm {tr} [F（a）-f（b） - （a-b）f'（b）] \ geq \ mathrm {tr} [f（| a-b |）] $$用于所有正矩阵$ a，b $。它尤其给出了克莱因对非负凸功能的不平等的改善。 结果，已经提出了詹森痕迹不平等的某些变体。
We show that if $f$ is a non-negative superquadratic function, then $A\mapsto\mathrm{Tr}f(A)$ is a superquadratic function on the matrix algebra. In particular, \begin{align*} \tr f\left( {\frac{A + B}{2}} \right) +\tr f\left(\left| {\frac{A - B}{2}}\right|\right) \leq \frac{{\tr {f\left( A \right)} + \tr {f\left( B \right)} }}{2} \end{align*} holds for all positive matrices $A,B$. In addition, we present a Klein's inequality for superquadratic functions as $$ \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 an improvement of the Klein's inequality for non-negative convex function. As a consequence, some variants of the Jensen trace inequality for superquadratic functions have been presented.
