Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convexcone $\mathcal{F}_+^{k:n}$ of all functions $f$ on an arbitrary interval$I\subseteq\mathbb{R}$ whose derivatives $f^{(j)}$ of orders $j=k-1,\dots,n$are nondecreasing is characterized in terms of extreme rays of the cone$\mathcal{F}_+^{k:n}$. A simple description of the convex cone dual to$\mathcal{F}_+^{k:n}$ is given. These results are useful in, and were motivatedby, applications in probability. In fact, the results are obtained in a moregeneral setting with certain generalized derivatives of $f$ of the $j$th orderin place of $f^{(j)}$. Somewhat similar results were previously obtained in thecase when the left endpoint of the interval $I$ is finite, with certainadditional integrability conditions; such conditions fail to hold in thementioned applications.
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机译：令$ n $和$ k $为非负整数，使得$ 1 \ le k \ le n + 1 $。所有函数$ f $的凸锥$ \ mathcal {F} _ + ^ {k：n} $在任意区间$ I \ subseteq \ mathbb {R} $上的导数$ f ^ {（j）} $ $ j = k-1，\ dots，n $的递减特征是圆锥体$ \ mathcal {F} _ + ^ {k：n} $的极端光线。给出了与\\ mathcal {F} _ + ^ {k：n} $对偶的凸锥的简单描述。这些结果在概率应用中很有用，并受其启发。实际上，结果是在更一般的情况下使用第f个$ j $阶的某些$ f $的广义导数代替$ f ^ {（（j）} $）获得的。在区间$ I $的左端点是有限的情况下，在某些附加可积性条件下，以前可获得类似的结果。这些条件在上述申请中均不成立。
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