  In recent papers the convexity of quasiarithmetic means was characterized
under twice differentiability assumptions. One of the main goals of this paper
is to show that the convexity or concavity of a quasiarithmetic mean implies
the the twice continuous differentiability of its generator. As a consequence
of this result, we can characterize those quasiarithmetic means which admit a
lower convex and upper concave quasiarithmetic envelop.
