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, ∞) → R is an operator midpoint superquadratic function if and only if f (C * AC) ≤ C * f (A)C − f C * A 2 C − (C * AC) 2 holds for every positive operator A ∈ B(H) + and every contraction C. As applications, some inequalities for quasi-arithmetic operator means are given