Increasing Convex Orders
The standard definition of increasing Convex Orders is:
Let X and Y be 2 r.v. such that E[f(X)] <= E[f(Y)] for all increasing convex [concave] functions f(.), THEN X is said to be smaller than Y in the increasing convex order.
My Question is:
Do you know whether it follows the other direction?
In other words, if X <= Y in the increasing convex order, does this imply that E[f(X)] <= E[f(Y)] for all increasing convex [concave] functions f(.)?
If the answer is YES, do you know any legitimate reference for me to use?
Thank you very much.