Inequalities involving operator superquadratic functions

Abstract

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 $f:[0,\infty)\to\mathbb{R}$ is an operator midpoint superquadratic function if and only if
\begin{align*}
f\left(C^*AC\right)\leq C^*f(A)C-f\left(\sqrt{C^*A^2C-(C^*AC)^2}\right)
\end{align*}
holds for every positive operator $A\in\mathcal{B}(\mathcal{H})^+$ and every contraction $C$.
As an application, some inequalities for quasi-arithmetic operator means are given.