We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Rasa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, $$q\in \mathbb N$$ . In particular, 1-monotone functions are nondecreasing functions, and 2-monotone functions are convex functions. In general, q-monotone functions on [0, 1], for $$q\ge 2$$ , possess a $$(q-2)$$ nd derivative in (0, 1), which is convex there. We also discuss some other linear positive approximation processes.

中文翻译：

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Raşa 猜想对 q-单调函数的扩展

我们扩展了一个涉及伯恩斯坦基多项式和 [0, 1] 上的凸函数的不等式。这个不等式最初是由 Rasa 在大约三十年前提出的，但直到最近才被证明。我们的扩展提供了一个涉及 q 单调函数的不等式， $$q\in \mathbb N$$ 。特别地，1-单调函数是非减函数，2-单调函数是凸函数。一般而言，[0, 1] 上的 q-单调函数，对于 $$q\ge 2$$ ，在 (0, 1) 中具有 $$(q-2)$$ nd 导数，该导数在那里是凸的。我们还讨论了其他一些线性正逼近过程。