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We present a tight parametrical Hermite-Hadamard type inequality withprobability measure, which yields a considerably closer upper bound for themean value of convex function than the classical one. Our inequality becomesequality not only with affine functions, but also with a family of V-shapedcurves determined by the parameter. The residual (error) of this inequality isstrictly smaller than in the classical Hermite-Hadamard inequality under anyprobability measure and with all non-affine convex functions. In the frameworkof Karamata's theorem on the inequalities with convex functions, we propose amethod of measuring a global performance of inequalities in terms of averageresiduals over functions of the type $x\mapsto |x-u|$. Using average residualsenables comparing two or more inequalities as themselves, with same ordifferent measures and without referring to a particular function. Our methodis applicable to all Karamata's type inequalities, with integrals or sums. Anumerical experiment with three different measures indicates that the averageresidual in our inequality is about 4 times smaller than in classical rightHermite-Hadamard, and also is smaller than in Jensen's inequality, with allthree measures.

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