  We observe that if f is a continuous function on an interval I and x_0 \in I,
then f is operator monotone if and only if the function (f(x) - f(x_0)/(x -
x_0) is strongly operator convex. Then starting with an operator monotone
function f_0, we construct a strongly operator convex function f_1, an
(ordinary) operator convex function f_2, and then a new operator monotone
function f_3. The process can be continued to obtain an infinite sequence which
cycles between the three classes of functions. We also describe two other
constructions, similar in spirit. We prove two lemmas which enable a treatment
of those aspects of strong operator convexity needed for this paper which is
more elementary than previous treatments. And we discuss the functions phi such
that the composite phi \circ f is operator convex or strongly operator convex
whenever f is strongly operator convex.
