  This paper concerns three classes of real-valued functions on intervals,
operator monotone functions, operator convex functions, and strongly operator
convex functions. Strongly operator convex functions were previously treated in
[3] and [4], where operator algebraic semicontinuity theory or operator theory
were substantially used. In this paper we provide an alternate treatment that
uses only operator inequalities (or even just matrix inequalities). We also
show that if t_0 is a point in the domain of a continuous function f, then f is
operator monotone if and only if (f(t) - f(t_0))/(t - t_0) is strongly operator
convex. Using this and previously known results, we provide some methods for
constructing new functions in one of the three classes from old ones. We also
include some discussion of completely monotone functions in this context and
some results on the operator convexity or strong operator convexity of phi
\circ f when f is operator convex or strongly operator convex.
