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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.21678 |
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| _version_ | 1866908287285329920 |
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| author | Páles, Zsolt Szostok, Tomasz |
| author_facet | Páles, Zsolt Szostok, Tomasz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_21678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orderings on measures induced by higher-order monotone functions Páles, Zsolt Szostok, Tomasz Classical Analysis and ODEs 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. |
| title | Orderings on measures induced by higher-order monotone functions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.21678 |