In this article we give a variant of the Hermite–Hadamard integral inequality for twice differentiable functions. It represents an improvement of this inequality in the case of convex/concave functions. Sharp two-sided inequalities for Simpson’s rule are also proven along with several extensions.

中文翻译：

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非凸函数的Simpson规则和Hermite-Hadamard不等式

在本文中，我们给出了两次可微函数的Hermite-Hadamard积分不等式的变体。在凸/凹函数的情况下，它代表了这种不等式的改善。辛普森法则的尖锐的两边不等式以及一些扩展也得到了证明。