  In [B1, Theorem 2.36] we proved the equivalence of six conditions on a
continuous function f on an interval. These conditions define a subset of the
set of operator convex functions, whose elements are called strongly operator
convex. Two of the six conditions involve operator-algebraic semicontinuity
theory, as given by C. Akemann and G. Pedersen in [AP], and the other four
conditions do not involve operator algebras at all. Two of these conditions are
operator inequalities, one is a global condition on f, and the fourth is an
integral representation of f stronger than the usual integral representation
for operator convex functions. The purpose of this paper is to make the
equivalence of these four conditions accessible to people who do not know
operator algebra theory as well as to operator algebraists who do not know the
semicontinuity theory. We also provide a similar treatment of one theorem from
[B1] concerning (usual) operator convex functions. And in two final sections we
give a somewhat tentative treatment of some other operator inequalities for
strongly operator convex functions, and we give a differential criterion for
strong operator convexity.
