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

DOI10.1007/S10998-021-00425-7zbMATH Open1513.26061arXiv2004.07567MaRDI QIDQ6338815

Zoran D. Mitrović, Milan J. Merkle

Publication date: 16 April 2020

Abstract: 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 $xmapsto|x-u|$. 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.

Mathematics Subject Classification ID

Inequalities; stochastic orderings (60E15) Inequalities for sums, series and integrals (26D15) Convexity of real functions in one variable, generalizations (26A51)

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