A tight Hermite–Hadamard inequality and a generic method for comparison between residuals of inequalities with convex functions

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.