Orderings on measures induced by higher-order monotone functions
Zsolt Páles, Tomasz Szostok
TL;DR
This work analyzes when the functional inequality $\int_{[0,1]} f((1-t)x+ty)\,d\mu(t) \ge 0$ holds for a continuous $f$ given a bounded signed measure $\mu$ on $[0,1]$, by connecting it to higher-order monotonicity notions (\(n\)-increasing/decreasing). It develops a framework based on divided differences and moment-like objects $\mu_k$, $\mu_k^-$, and $\mu_k^+$, including a mollification smoothing result to handle non-smooth $f$, and leverages Popoviciu characterizations to obtain a sufficiency theorem: if $\mu_k(\lambda)\,f$ is $k$-increasing for $k< n$ and $f$ has compatible $n$-order monotonicity with appropriate sign conditions on $\mu_{n-1}^-$ and $\mu_{n-1}^+$, then the inequality holds for all $x<y$. The paper also derives necessary conditions via test functions and presents numerous examples showing how different combinations of monotone orders yield admissible function classes, thereby extending stochastic ordering results beyond classical convex orderings. Overall, it provides a flexible, constructive method to obtain function classes satisfying linear functional inequalities induced by measures, with implications for stochastic ordering and higher-order convexity concepts.
Abstract
The main aim of this paper is to study the functional inequality \begin{equation*} \int_{[0,1]}f\bigl((1-t)x+ty\bigr)dμ(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 $μ$ 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.