论文标题
紧密的Hermite-Hadamard的不平等和一种通用方法，用于比较具有凸功能的不平等的残差
A tight Hermite-Hadamard's inequality and a generic method for comparison between residuals of inequalities with convex functions
论文作者
论文摘要
我们提出了具有概率度量的紧密参数hermite-hadamard类型的不等式，这比经典函数的平均值更接近上限。我们的不平等不仅在仿射功能上变得平等，而且与由参数确定的V形曲线的家族一样。这种不平等的残差（误差）严格小于在任何概率措施和所有非携带凸函数下的经典Hermite-Hadamard不平等现象中。在Karamata关于凸功能不平等的定理的框架中，我们提出了一种方法，以平均残留物在$ x \ mapsto | x-u | $的类型功能方面，以平均残差为角度来衡量不平等的全球性能。使用平均残留物可以比较两个或多个不平等，并以相同或不同的度量进行比较，而无需指代特定功能。我们的方法适用于所有Karamata的类型不等式，具有积分或总和。一个具有三种不同措施的数值实验表明，我们不平等中的平均残差比经典右Hermite-Hadamard小的4倍，并且比Jensen的不平等中的平均残差和所有三种措施都小。
We present a tight parametrical Hermite-Hadamard type inequality with probability measure, which yields a considerably closer upper bound for the mean value of convex function than the classical one. Our inequality becomes equality not only with affine functions, but also with a family of V-shaped curves determined by the parameter. The residual (error) of this inequality is strictly smaller than in the classical Hermite-Hadamard inequality under any probability measure and with all non-affine convex functions. In the framework of Karamata's theorem on the inequalities with convex functions, we propose a method of measuring a global performance of inequalities in terms of average residuals over functions of the type $x\mapsto |x-u|$. Using average residuals enables comparing two or more inequalities as themselves, with same or different measures and without referring to a particular function. Our method is applicable to all Karamata's type inequalities, with integrals or sums. A numerical experiment with three different measures indicates that the average residual in our inequality is about 4 times smaller than in classical right Hermite-Hadamard, and also is smaller than in Jensen's inequality, with all three measures.
