  The new notion of operator/matrix $k$-tone functions is introduced, which is
a higher order extension of operator/matrix monotone and convex functions.
Differential properties of matrix $k$-tone functions are shown.
Characterizations, properties, and examples of operator $k$-tone functions are
presented. In particular, integral representations of operator $k$-tone
functions are given, generalizing familiar representations of operator monotone
and convex functions.
