Inequalities involving operator superquadratic functions

In this paper, related to the well-known operator convex functions, we study a class of operator functions, the operator superquadratic functions. We present some Jensen-type operator inequalities for these functions. In particular, we show that $f:[0, infty)tomathbb{; ; R}; ; $ is an operator midpoint superquadratic function if and only if $ fleft(C^*ACright)leq C^*f(A)C- fleft(sqrt{; ; C^*A^2C-(C^*AC)^2}; ; right)$ holds for every positive operator $Ainmathcal{; ; B}; ; (mathcal{; ; H}; ; )^+$ and every contraction $C$. As an application, some inequalities for quasi- arithmetic operator means are given.