Orderings on measures induced by higher-order monotone functions

The main aim of this paper is to study the functional inequality \begin{equation*} \int_{[0,1]}f\bigl((1-t)x+ty\bigr)d\mu(t)\geq 0, \qquad x,y\in I \mbox{ with } x<y, \end{equation*} for a continuous unknown function $f:I\to{\mathbb R}$, where $I$ is a nonempty open real interval and $\mu$ is a signed and bounded Borel measure on $[0,1]$. We derive necessary as well as sufficient conditions for its validity in terms of higher-order monotonicity properties of $f$. Using the results so obtained we can derive sufficient conditions under which the inequality $${\mathbb E} f(X)\leq {\mathbb E} f(Y)$$ is satisfied by all functions which are simultaneously: $k_1$-increasing (or decreasing), $k_2$-increasing (or decreasing), \dots , $k_l$-increasing (or decreasing) for given nonnegative integers $k_1,\dots,k_l.$ This extends several well-known results on stochastic ordering. A necessary condition for the $(n,n+1,\dots,m)$-increasing ordering is also presented.