How to prove that for two measures with finite moments,
$$ \forall x \in \mathbb R, \quad \int |y-x| d \mu (y) \leq \int |y-x| d \nu (y)$$
$ \qquad \qquad \implies \forall \phi, \text{ convex function, we have that }$
$$\int \phi(y) d \mu (y) \leq \int \phi(y) d \nu (y) $$
Perhaps there should...
> In French, an increasing process for the convex order reads: un Processus Croissant pour l'Ordre Convexe, which yields the acronym PCOC. This being pronounced “peacock”, we adopt this name for such processes.
From Peacocks and Associated Martingales, with Explicit Constructions, page 13.
