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.
Categories: math.FA