Convex order and an integral inequality over quantile functions Let $X, Y$ be random variables in $L^1$. We will write $X \preceq_c Y$ if for all convex functions $u$ on $\mathbb{R}$
$$ \mathbb{E} [u(X)] \leq \mathbb{E}[u(Y)].$$
Now let $q_X$ and $q_Y$ be the (left-continuous) quantile functions of $X$ and $Y$.
How to prove that
$$ \int_0^t q_X(s) d s \geq \int_0^t q_Y(s) ds$$
for all $0 \leq t \leq 1$ given $X \preceq_c Y$?
The case $t=0$ is trivial. For $t=1$ note that if $U$ is uniformly distributed on $(0,1)$, then $q_X (U)$ is distributed as $X$. Thus,
$$\int_0^1 q_X (s) ds = \mathbb{E}[q_X(U)] = \mathbb{E} [X] = \mathbb{E}[Y] = \int_0^1 q_Y (s) ds $$
since $x \mapsto x$ and $x \mapsto -x$ are both convex.
But I don't see how this can be generalized to prove the case where $t$ is not $0$ or $1$.
 A: I think you are trying to prove something like Theorem 5 in David Blackwell's paper Comparison of Experiments .  Blackwell's functions $F_M$ and $F_m$ are, I think, the inverse functions of your $q_X$ and $q_Y$ (or maybe vice versa; the notations make my head spin).  In general, your hypothesis $Eu(X)\le Eu(Y)$ for all convex $u$ is pretty strong, and has well-known necessary and sufficient equivalents, as in the famous paper of Cartier, et al.  The exposition in Le Cam's Comparison of Experiments - A short review (see esp. the bottom of page 130) might serve as a roadmap.
A: It is relatively well-known that $X \preceq_c Y$ if and only if $Y$ is a mean-preserving spread of $X$. (If this is not something you already know, I imagine it might be found in the links in kimchi lover's answer. Otherwise, it's probably in Shaked and Shanthikumar.)
The result you are looking for is stated as lemma $1$ of Brooks and Du (2018). The proof is in their appendix.
Apologies for the brief answer. I don't have time to write a detailed one at the moment, and I thought you'd rather a brief one sooner rather than a detailed one (possibly much) later. I'll try to find the time to develop this into a full answer later on.
A: I found a source for a direct proof. Thanks to the two answers for telling me where to look.
