  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.
