  In this work, an operator superquadratic function (in operator sense) for
positive Hilbert space operators is defined. Several examples with some
important properties together with some observations which are related to the
operator convexity are pointed out. Equivalent statements of a non-commutative
version of Jensen's inequality for operator superquadratic function are
established. A generalization of the main result to any positive unital linear
map is also provided.
