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 (a distance between two sides) of this inequality is strictly smaller than in the classical Hermite–Hadamard inequality under any probability measure and with all nonaffine 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. Some topics from history and priority are discussed.

我们提出了一个带有概率测度的严格参数 Hermite-Hadamard 型不等式，它产生了比经典函数更接近凸函数平均值的上限。我们的不等式不仅与仿射函数相等，而且与由参数确定的一系列 V 形曲线相等。该不等式的残差（两侧之间的距离）在任何概率测度和所有非仿射凸函数下都严格小于经典 Hermite-Hadamard 不等式。在关于具有凸函数的不等式的 Karamata 定理的框架中，我们提出了一种根据类型\(x\mapsto |xu|\) 的函数的平均残差来衡量不等式的全局性能的方法. 使用平均残差可以比较两个或多个不等式本身，使用相同或不同的度量并且不涉及特定函数。我们的方法适用于所有带有积分或和的 Karamata 类型不等式。使用三种不同测度的数值实验表明，我们不等式中的平均残差比经典右 Hermite-Hadamard 小约 4 倍，也小于使用所有三种测度的 Jensen 不等式。讨论了历史和优先事项中的一些主题。