Interchanging expecation and a monotone function Are there any theorems similar to Jensen's Inequality for monotone functions instead of convex functions? For example, if I know $EX \le EY$, can I say anything about $Ef(X)$ and $Ef(Y)$ given that I know $f$ is a nondecreasing function?

If $X$ and $Y$ are stochastically ordered, i.e. say $X\leq_{st.}Y$, then we have both $$\mathbb{E}(X)\leq \mathbb{E}Y\ \mbox{and}\ \mathbb{E}(f(X))\leq \mathbb{E}(f(Y))$$ where $f$ is a non-decreasing function.

Conversely, if for any non-decreasing function $f$ we have $$\mathbb{E}(f(X))\leq \mathbb{E}(f(Y))$$ then it follows that $X\leq_{st.}Y$ and consequently $\mathbb{E}(X)\leq \mathbb{E}(Y)$.

For more on stochastic ordering see here.